Delta Function Research Articles

Nonlocal Traffic Models with General Kernels: Singular Limit, Entropy Admissibility, and Convergence Rate

Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counter-examples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement on the convolution kernels. We then provide a general criterion for entropy admissibility of the limit and a convergence rate. We also exhibit a counter-example showing that the convexity assumption is necessary for our main compactness estimate.

A point particle source model for conjugate heat and mass transfer in dispersed two-phase flows by BEM based methods

The paper reports on development of a Boundary Element Method (BEM) based two phase dispersed flow model, that allows an accurate and efficient heat and mass transfer coupling by using a moving point source model to account for particle–fluid interaction. The two-way coupling model highlights the efficient use of the elliptic fundamental solution and the Dirac delta distribution properties to accurately evaluate the heat and mass point particle source impact on the continuous phase, solved by the Boundary Domain Integral Method (BDIM). In addition to the BDIM model of the particle–fluid heat and mass transfer interaction, the two-phase flow case under consideration is extended to the case of porous spherical particle drying with internal moving drying front, which is solved by the Boundary Element Method. As the two-phase flow is considered to be dilute, the particle–fluid momentum exchange is covered by a one-way coupling algorithm, with Lagrangian particle tracking used for determination of particle positions and velocities in the flow. Two computational cases are presented, where 1000 and 10000 particles are dried in a stream of hot air. Comparison between the obtained drying times for the cases of the one-way and the two-way heat and mass transfer coupling results shows, that the developed two-way coupling model accurately captures the effect of moisture accumulation and temperature decrease in the fluid phase, leading to realistic computations of drying rates of particles in the flow.

Holographic p-Wave Superconductor with Excited States in 4D Einstein–Gauss–Bonnet Gravity

We construct a holographic p-wave superconductor with excited states in the 4D Einstein–Gauss–Bonnet gravity using the Maxwell complex vector field model. In the probe limit, we observe that, the higher curvature correction or the higher excited state can hinder the vector condensate to be formed in the full parameter space, which is different from the holographic s-wave superconductor. Regardless of the choice of the vector mass by selecting the value of m2L2 or m2Leff2, we note that the critical chemical potential becomes evenly spaced for the number of nodes and that the difference of the critical chemical potential between the consecutive states depends on the curvature correction. Moreover, we find that the higher curvature correction or the higher excited state will alter the universal relation of the gap frequency, and the pole and delta function of the conductivity for the excited states can be broadened into the peaks with the finite width as the curvature correction increases.

Dynamic response of a damaged bridge model traversed by a heavy point mass

The dynamic response of a single-span, damaged bridge model traversed by a point mass whose magnitude is comparable to that of the underlying girder is studied for the purpose of detecting structural damage. Damage is represented in the form of vertical cracks running across the girder's web, which modifies the eigenproperties of the original intact girder. The weight of the mass during its traverse time also modifies the eigenproperties of the girder and its motion produces an additional damping effect. Both features are modelled as Dirac delta distributions, one accounting for the spatial discontinuity in the slope of the deflection, and the other for the temporal discontinuity as the point mass moves forward with constant velocity. Although each type of discontinuity has been treated in the past, their combined effect has not been satisfactorily addressed. Next, a novel time-marching scheme is introduced for treating the temporal discontinuity, where the mass, stiffness and damping of the girder are all updated at the end of a given time interval. Then, the equations of motion are cast as a first-order, matrix differential equation. A parametric study has shown that two eigenvalue-eigenfunction pairs are sufficient for convergence purposes. Following modal synthesis, the transient velocities at any point on the girder during both forced and free vibration regimes are processed by the short-time Fourier transform as a means for detecting changes in the resulting spectograms that are attributable to damage. Finally, support failure is also examined by the introduction of flexible translational springs at the ends. The methodology developed here can be used to identify damage from the recorded kinematic spectrograms at any station on the span of the girder due to a travelling mass. In closing, this work finds applications in structural health monitoring of structures, which is a contemporary and challenging problem facing the engineering profession.

Renormalization in a wavelet basis

Discrete wavelet-based methods promise to emerge as an excellent framework for the non-perturbative analysis of quantum field theories. In this work, we investigate aspects of renormalization in theories analyzed using wavelet-based methods. We demonstrate the nonperturbative approach of regularization, renormalization, and the emergence of flowing coupling constant within the context of these methods. This is tested on a model of the particle in an attractive Dirac delta function potential in two spatial dimensions, which is known to demonstrate quintessential features found in a typical relativistic quantum field theory.

On the Exponential Decay of Strongly Interacting Cold Atoms from a Double-Well Potential

In this article, we study an exponential decay for the gas of bosons with strong repulsive delta interactions from a double-well potential. We consider an exactly solvable model comprising an infinite wall and two Dirac delta barriers. We explore its features both within the exact method and with the resonance expansion approach. The study reveals the effect of the splitting barrier on the decay rate in dependence on the number of particles. Among other things, we find that the effect of the splitting barrier on the decay rate is most pronounced in systems with odd particle numbers. During exponential decay, the spatial correlations in an internal region are well captured by the “radiating state”.

Numerical simulation and experimental research of electrospun polyacrylonitrile Taylor cone based on multiphysics coupling

Abstract In the electrospinning process, the Taylor cone, as the jet source, directly affects the jet movement and the quality of the fiber membrane. Therefore, to understand the formation mechanism of the Taylor cone intuitively, a multiphysics coupling model that comprehensively considers the gravitational field, electrostatic field, and fluid field is established, and numerical simulations are conducted in this study. First, we construct a level-set function and analyze the force of the droplet. The gravity, surface tension, and electric field force are coupled to the incompressible Navier–Stokes equation as volume forces, and the nonconservation of the droplet area is solved by approximating the Dirac function with a smooth function. Subsequently, the deformation of the electrospun polyacrylonitrile (PAN) Taylor cone under different process parameters is simulated. Finally, data obtained from the numerical simulation and the average diameter of the electrospun PAN fiber membrane are analyzed via gray relational analysis. The results show that the volume force is the key factor affecting the average diameter of the fiber membrane (the correlation is 0.934). This article provides an effective reference and basis for the analysis and control of the electrospinning process.

The global well‐posedness for nonlinear Schrödinger equation with the delta potential in the energy space

In this paper, we concern with the global well‐posedness for nonlinear Schrödinger equation with a Dirac delta potential ( ‐NLS): Firstly, we give a new variational characterization of the solitary waves of ( ‐NLS) by making use of the symmetric decreasing rearrangement and constrained minimization argument. Secondly, we construct an invariant set , via the variational characterization of the solitary waves. Finally, we show the solutions, with initial data in , exist globally.

An Explanation for a Paradox in a Fluid-Discharging Cantilevered Pipe Attached With an End-Mass

Abstract For a fluid-discharging cantilevered pipe attached with an end-mass, there are two methods to account for the end-mass effect. The first is that the end-mass is considered in the boundary conditions. The second is that the end-mass is included in the equation of motion via a Dirac delta function. As the analytical solution of the linear free vibration model is not available due to the presence of Coriolis force, the eigenfunctions of a beam, which satisfy the same boundary conditions, are commonly employed in the Galerkin method. It has been found the first method is incorrect for natural frequency calculation when the internal flow velocity is nonzero. However, the intrinsic mechanism remains to be clarified. This study has demonstrated the eigenfunctions in the first method depend on the end-mass and the orthogonality relations are quite different from that of typical simple beams, based on which a new model is proposed and the prediction compares well with that in the second method. For further validation, the critical internal flow velocity, the onset flutter frequency, and the dynamic responses of suspended pipes under gravity are computed, which compare well with experimental observations. This study can provide a workbench for fluid-conveying pipes with various boundary conditions.

On the image method of separable variable problems

The image method is a classical analysis method for mathematical physics problems but restricted by domain geometry and boundary conditions. In this paper, the scope of the image method is investigated and extended to separable variable problems. The general description of the image method is established and the image source distribution is determined for separable variable problems. The image source distribution function is proved to be the out of domain part of the eigenfunction series of the source distribution function. This continuation from the source distribution function to the image source distribution function is also given as the inner product with the eigenfunction series of the Dirac delta function. Three numerical examples with complex boundary conditions or geometry domains are prepared to verify the generality and effectiveness of the conclusion.

Meshless method and experimental analysis of hyperelastic plates using Mooney-Rivlin strain energy function subjected to concentrated loading

The purpose of this research is to investigate a strong-form meshless method for experimentally and numerically analyzing hyperelastic plates subjected to concentrated loading. The silicone hyperelastic plate was formulated utilizing Mooney-Rivlin strain energy function and Lagrangian strains. The displacements and Lagrangian strains of the plate are also calculated using the first-order shear deformation plate theory (FSDPT). The hyperelastic plate’s governing equations are deduced in strong form and analyzed using the logarithmic thin-plate spline radial basis function (LTPS-RBF) and the meshless method for the first-time. The nonlinear system of governing equations is solved using the arc-length continuation algorithm. In the meshless method, the concentrated loading is modeled using the Kronecker delta function in the expression of external work. A silicone plate is employed experimentally to evaluate the accuracy of the meshless method’s results. Furthermore, the finite element method is utilized to compare the contours obtained from the meshless method. Concentrated loading is considered at various points to validate the meshless methods and results from the experimental analysis. The most significant difference between the meshless method and the experimental results is less than 15%. The results indicate that the meshless method is highly accurate when analyzing hyperelastic plates subjected to concentrated loading.

Solute transport in fractured porous media: Implementation of the time domain random walk algorithm for different injection boundaries

In this work, three variant Time Domain Random Walk (TDRW) algorithms were developed for the problem of solute transport in a single fracture-matrix system, where an arbitrary inlet boundary condition can be applied. One approach performs an additional evaluation of integral in terms of injection boundary and the solution to a Dirac delta case. One method makes use of two functions, dependent on specific boundary conditions, to estimate the particle arrival time. The other additionally introduces the concept of solute injection time, resulting from the injection boundary, into the calculation of particle arrival time. To validate the developed variant algorithms, two benchmark cases are considered with respect to a general Dirichlet injection mode and a Robin injection boundary, respectively. The results from three approaches all make a good agreement with those of inverse Laplace transform method. However, the Monte Carlo nature of the TDRW algorithm implies that the accuracy of the computational result is highly dependent on the number of particles applied in the simulation.

Students’ difficulties with the Dirac delta function in quantum mechanics

In the context of quantum mechanics, students are often asked to use delta functions to solve problems. Here, we investigate three typical problem-solving processes using delta functions: a delta function potential well problem, a position space delta function problem, and a momentum space delta function problem. We studied students’ solutions in written exams and conducted think-aloud interviews. We use the activation, construction, execution, and reflection framework for our analysis. We find that students encountered various difficulties in solving quantum mechanics problems using delta functions. Common challenges included difficulty with establishing expressions of delta functions for position eigenstates, difficulty with expressing the orthonormality of eigenfunctions with continuous spectra using delta functions, difficulty with boundary conditions for delta function potentials, and difficulty with calculating integrals involved delta functions. In particular, students rarely used effective reflective methods to gain confidence in their solutions. We commonly observed this challenge for students in all questions we investigated. In addition, we compare the similarities and differences in the use of delta functions in electrostatics and in quantum mechanics, and discuss possible explanations for the reasoning mechanisms that cause these difficulties. Finally, we discuss the potential pedagogical implications of our findings.Received 26 May 2022Accepted 23 January 2023DOI:https://doi.org/10.1103/PhysRevPhysEducRes.19.010104Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasEpistemology, attitudes, & beliefsLearning theoryScientific reasoning & problem solvingPhysics Education Research

Multi‐task multi‐scale attention learning‐based facial age estimation

Face-based age estimation strongly depends on deep residual networks (ResNets), used as the backbone in the relevant research. However, ResNet-based methods ignore the importance of some large-scale facial information and other facial age attributes. Inspired by the attention mechanism, a multi-task learning framework for face-based age estimation called multi-task multi-scale attention is proposed. First, the authors embed the alternative strategy structure of dilated convolution into ResNet34 to construct a multi-scale attention module (MSA) to improve the network's receptive field, which extracts local age-sensitive information while obtaining multi-scale features. The MSA can have a larger receptive field to extract both large-scale and local detailed feature information. Second, multi-task learning network structures are built to predict gender and race, which can share rigid network parameters to improve age estimation and improve the accuracy of age estimation by other age-related parameters. Finally, the Kullback-Leibler divergence loss is adopted between a Dirac delta label and a Gaussian prediction to guide the training. The numerical tests on the MORPH Album II and Adience datasets prove the superiority of the proposed method over other state-of-the-art ones.

On spectral properties of the Schreier graphs of the Thompson group 𝐹

In this article we study spectral properties of the family of Schreier graphs associated to the action of the Thompson group F F on the interval [ 0 , 1 ] [0,1] . In particular, we describe spectra of Laplace type operators associated to these Schreier graphs and calculate certain spectral measures associated to the Schreier graph Υ \Upsilon of the orbit of 1 / 2 1/2 . As a byproduct we calculate the asymptotics of the return probabilities of the simple random walk on Υ \Upsilon starting at 1 / 2 1/2 . In addition, given a Laplace type operator L L on a tree-like graph we study relations between the spectral measures of L L associated to delta functions of different vertices and the spectrum of L L .

Competing spin-valley entangled and broken symmetry states in the N=1 Landau level of graphene

We study many-body ground states for the partial integer fillings of the $N=1$ Landau level in graphene, by constructing a model that accounts for the lattice scale corrections to the Coulomb interactions. Interestingly, in contrast to the $N=0$ Landau level, this model contains not only pure delta function interactions but also some of its derivatives. Due to this we find several important differences with respect to the $N=0$ Landau level. For example at quarter filling when only a single component is filled, there is a degeneracy lifting of the quantum hall ferromagnets and ground states with entangled spin and valley degrees of freedom can become favourable. Moreover at half-filling of the $N=1$ Landau level, we have found a new phase that is absent in the $N=0$ Landau level, that combines characteristics of the Kekul\'{e} state and an antiferromagnet. We also find that according to the parameters extracted in a recent experiment, at half-filling of the $N=1$ Landau level graphene is expected to be in a delicate competition between an AF and a CDW state, but we also discuss why the models for these recent experiments might be missing some important terms.

Continuous modeling of creased annuli with tunable bistable and looping behaviors

Creases are purposely introduced to thin structures for designing deployable origami, artistic geometries, and functional structures with tunable nonlinear mechanics. Modeling the mechanics of creased structures is challenging because creases introduce geometric discontinuity and often have complex mechanical responses due to local material damage. In this work, we propose a continuous description of the sharp geometry of creases and apply it to the study of creased annuli, made by introducing radial creases to annular strips with the creases annealed to behave elastically. We find that creased annuli have generic bistability and can be folded into various compact shapes, depending on the crease pattern and the overcurvature of the flat annulus. We use a regularized Dirac delta function (RDDF) to describe the geometry of a crease, with the finite spike of the RDDF capturing the localized curvature. Together with anisotropic rod theory, we solve the nonlinear mechanics of creased annuli, with its stability determined by the standard conjugate point test. We find excellent agreement between precision tabletop models, numerical predictions from our analytical framework, and modeling results from finite element simulations. We further show that by varying the rest curvature of the thin strip, dynamic switches between different states of creased annuli can be achieved, which could inspire the design of deployable and morphable structures. We believe that our smooth description of discontinuous geometries will benefit the mechanical modeling and design of a wide spectrum of engineering structures that embrace geometric and material discontinuities.

On An Illustrative Examples of a Studied Noetherity Dirac-Delta Extensions for a Noether Operator

The purpose of this work is to illustrate by clear examples the noetherity nature of a finite Dirac-delta Extensions of a studied noether operator. Previously in our published papers, we have investigated in different two cases, the noetherization of a Dirac-delta extensions of a noether linear integro-differential operator defined by a third kind integral equation in some specific well chosen functional spaces. Our various already published researches were connected with such topic widely studied and clearly presenting different specific approaches, applied when establishing fundamentaly noether theory for some kind of integro-differential operators to reach the noetherization. The initial considered noether operator <I>A</I> has been extended with some finite dimensional spaces of Dirac-delta functions, and the noetherization of the two cases of extensions has been established depending with the parameters of the third kind integral equation defining <I>A</I>. The previous lead us to set the problem of the construction of practical examples clearly illustrating the relationship between theory and practise. For this aim, we based on an established wellknown noether theory and, we construct in this work step by step, two illustrative examples to show the interconnexion between the theory and pratise related to the investigation of the construction of noether theory for the considered extended noether operator denoted <img width="8" height="10" src="http://article.sciencepublishinggroup.com/journal/347/3471192/image001.png" /> defined by a third kind linear singular integral equation in some generalized functional spaces. The extended operator <I>A</I> of the initial noether operator <I>A</I> is verified being also noether and therefore we deduce the index of the extended operator <img width="8" height="10" src="http://article.sciencepublishinggroup.com/journal/347/3471192/image001.png" />.

Multiple Exciton Generation Solar Cells: Numerical Approaches of Quantum Yield Extraction and Its Limiting Efficiencies

Multiple exciton generation solar cells exhibit low power conversion efficiency owing to non-radiative recombination, even after the generation of numerous electron and hole pairs per incident photon. This paper elucidates the non-idealities of multiple exciton generation solar cells. Accordingly, we present mathematical approaches for determining the quantum yield to discuss the non-idealities of multiple exciton generation solar cells by adjusting the delta function. We present the use of the Gaussian distribution function to present the occupancy status of carriers at each energy state using the Dirac delta function. Further, we obtained ideal and non-ideal quantum yields by modifying the Gaussian distribution function for each energy state. On the basis of this approach, we discuss the material imperfections of multiple exciton generations by analyzing the mathematically obtained quantum yields. Then, we discuss the status of radiative recombination calculated from the ratio of radiative to non-radiative recombination. Finally, we present the application of this approach to the detailed balance limit of the multiple exciton generation solar cell to evaluate the practical limit of multiple exciton generation solar cells.

A Verification of Riemann Non-Trivial Zeros by Complex Analysis by Matlab™ Computation

With my most recent paper, I tried to prove the Riemann Hypothesis by catching out those contradictory parts of the non-trivial zeros. In the present paper, I will try to verify these known values of Riemann nontrivial zeros by first using U.S.A. Matlab coding with a list of well-organized complex analysis theories. At the same time, as the major core of my verification is just a mono-direction one (i.e. there may be a possibility of the missing non-trivial zeros although the residue value is zero), hence this author try to solve such problem by assuming that there are some other zeros existing between the two known zeros but the contradiction arises – as singularity implies the residue has a value with a multiple of 2πi. In addition, this author also apply the ingenious design (or a hybrid skill) with Feynman technique and Integration by parts to solve a special zeta function integral. Next, this author finds that one may consider those non-trivial zeros as a Fourier transform (or an impulse) between other normal complex numbers. The result is consistent with my previous papers in quantum physics [23], [25] for the electron jumps or reverse. Hence, we may get the (dirac) delta equation for Riemann Zeta. Then we may formulate our quantum circuit &amp; computer. Finally, this author concludes all findings with an algorithm for searching, finer and checking the non-trivial zeros like below: Step 1: Use the computer software with some suitable program codes for an elementary search of feasible non-trivial zeta values among the closed real-complex plane interval – Method Matlab Simulation for searching zeta zeros; Step 2: Substitute back the values laying in the contour interval for zeta as found in Step 1 into the limit of ln(zeta(z))/((zeta'(z)) ) in order to adjust the answer in a finer and accurate way (just like the case of Newton’s method etc) with more decimal digitals – Method Ingenious Design for finer the zeta zero’s values; Step 3: Employ the Cauchy Residue Theorem for a check and hence confirm the previous found non-trivial zeta roots’ uniqueness without any zeta zeros laying in between the two consecutive zeta roots – Method Cauchy’s Residue for checking those already found zeta zeros.