Delta Function Research Articles

Linear Quadratic Regulator for Delayed Systems Using the Hamiltonian Approach and Exact Closed-Loop Poles for First-Order Systems

Abstract We consider the linear quadratic regulator (LQR) for linear constant-coefficient delay differential equations (DDEs) with multiple delays. The Hamiltonian approach is used instead of an algebraic Riccati partial differential equation. Two coupled DDEs governing the state and control input are derived using the calculus of variations. This coupled system, with infinitely many roots in both left and right half-planes, defines a boundary value problem. Its left half-plane roots are the exact closed-loop poles of the controlled system. These closed-loop poles have not been used to compute the optimal feedback before. Here, the distributed delay kernel that yields exactly those poles is first computed using an eigenfunction expansion. Increasing the number of terms in the truncated expansion yields a highly oscillatory kernel. However, the oscillatory kernel's antiderivative converges to a piecewise smooth function on the delay interval plus a Dirac delta function at zero. Discontinuities in the kernel coincide with discrete delay values in the original DDE. Using this insight, a fitted piecewise polynomial kernel matches the exact closed-loop poles very well. The twofold contribution of the Hamiltonian approach is thus clarity on the form of the feedback kernel as well as the exact closed-loop poles. Subsequently, the fitted piecewise polynomial kernel can be used for a much simpler control calculation. The polynomial coefficients can be fitted by solving a few simultaneous linear equations. Two detailed numerical examples of the LQR for DDEs, one with two delays and one with three delays, show excellent results.

Topological Subordination in Quantum Mechanics

An example of non-Markovian quantum dynamics is considered in the framework of a geometrical (topological) subordination approach. The specific property of the model is that it coincides exactly with the fractional diffusion equation, which describes the geometric Brownian motion on combs. Both classical diffusion and quantum dynamics are described using the dilatation operator xddx. Two examples of geometrical subordinators are considered. The first one is the Gaussian function, which is due to the comb geometry. For the quantum consideration with a specific choice of the initial conditions, it corresponds to the integral representation of the Mittag–Leffler function by means of the subordination integral. The second subordinator is the Dirac delta function, which results from the memory kernels that define the fractional time derivatives in the fractional diffusion equation.

Complete \({\boldsymbol{SE(3)}}\) Invariants for a Comeagre Set of \({\boldsymbol{C^3}}\) Compact Orientable Surfaces in \(\mathbb{R}^{\boldsymbol{3}}\)

We introduce invariants for compact $C^1$-orientable surfaces (with boundary) in $\mathbb{R}^3$ up to rigid transformations. Our invariants are certain degree four polynomials in the moments of the delta function of the surface. We give an effective and numerically stable inversion algorithm for retrieving the surface from the invariants, which works on a comeagre subset of $C^3$-surfaces.

Reflection Loss is a Parameter for Film, not Material

In studies of microwave absorption in the current literature, theories such as reflection loss, impedance matching, the delta function, and the quarter-wavelength model have been inappropriately applied. As shown in this case study, these problems need to be corrected as they are representative of similar work in the literature.

Stability analysis for internal flow-induced vertical cantilevered pipe subject to multiple lumped masses

This study investigates the effects of both single and multiple lumped masses attached to a vertical cantilevered pipe conveying internal flow. By introducing delta and Heaviside functions, the governing equation of the vibration of a pipe with lumped masses is newly established based on Hamilton’s principle. The gravitational force of the lumped mass should be considered in the theoretical model but has been essentially ignored in previous research. The harmonic differential quadrature (HDQ) method is applied to solve the governing vibration equation. The newly developed model is validated through a comparison with published data. Numerical results show that the gravity force of the lumped mass plays an important role, and ignoring this force leads to inaccurate results in the investigation of lumped mass effects. The instability of the pipe is analyzed using this new model with various lumped mass weights, positions, and fluid mass ratios. The numerical results showed that lumped masses significantly affect the critical flow velocity, vibration frequency, and modal shapes. The conclusions are expected to supply valuable guidance to reduce pipe vibration using lumped masses in engineering applications.

Nonlocalization of singular potentials in quantum dynamics

Nonlocal modeling has drawn more and more attention and becomes steadily more powerful in scientific computing. In this paper, we demonstrate the superiority of a first-principle nonlocal model—Wigner function—in treating singular potentials which are often used to model the interaction between point charges in quantum science. The nonlocal nature of the Wigner equation is fully exploited to convert the singular potential into the Wigner kernel with weak or even no singularity, and thus highly accurate numerical approximations are achievable, which are hardly designed when the singular potential is taken into account in the local Schrödinger equation. The Dirac delta function, the logarithmic, and the inverse power potentials are considered. Numerically converged Wigner functions under all these singular potentials are obtained with a fourth-order accurate operator splitting spectral method, and display many interesting quantum behaviors as well.

Analysis of absorbing boundary conditions for the anomalous diffusion in comb model on unbounded domain by finite volume method

The diffusion in the comb model is an important kind of anomalous diffusion which is described by a governing equation containing the Dirac delta function and the solution’s domain is infinite. Two key problems are solved to analyze the mass transfer mechanism in the comb model. One is to construct the appropriate and reasonable boundaries for treating the infinite boundaries. The exact absorbing boundary conditions with the Caputo’s fractional derivative are deduced by the Laplace transform technique. The other one is applying the finite volume method by integrating the governing equation over a governed volume to obtain the numerical solution with the advantage that the effects of the singular Dirac functions disappear. In addition, by introducing a source term, the exact solution is deduced and the comparison between the numerical solution and the exact solution is presented, which verifies the effectiveness and reasonableness of the numerical method. The particle distributions with different parameters are analyzed and discussed. By comparing with the traditional truncated boundary method, it is noteworthy that the mean square displacement for the absorbing boundary conditions has a good agreement with the exact expression.

Holographic superconductors at zero density

We construct holographic superconductors at zero density. The model enjoys a luxury property that the background geometry dual to the ground state is analytically available. It has a hyperscaling-violating geometry in the IR and is asymptotically AdS in the UV. Classification by IR geometries gives new insights on supergravity solutions. We numerically construct the finite temperature solution of hairy black holes and verify the phase transition by tuning a double-trace deformation parameter. For a holographic superconductor from M-theory, we obtain an analytic solution of the AC conductivity, which explicitly shows a superconducting delta function and a hard gap.

Stochastic analysis for vector-valued generalized grey Brownian motion

In this article, we show that the standard vector-valued generalization of a generalized grey Brownian motion (ggBm) has independent components if and only if it is a fractional Brownian motion. In order to extend ggBm with independent components, we introduce a vector-valued generalized grey Brownian motion (vggBm). The characteristic function of the corresponding measure is introduced as the product of the characteristic functions of the one-dimensional case. We show that for this measure, the Appell system and a calculus of generalized functions or distributions are accessible. We characterize these distributions with suitable transformations and give a d d -dimensional Donsker’s delta function as an example for such distributions. From there, we show the existence of local times and self-intersection local times of vggBm as distributions under some constraints, and compute their corresponding generalized expectations. At the end, we solve a system of linear SDEs driven by a vggBm noise in d d dimensions.

Security-Guaranteed PID Control for Discrete-Time Systems Subject to Periodic Dos Attacks

This paper is concerned with the observer-based H∞ proportional-integral-derivative (PID) control issue for discrete-time systems using event-triggered mechanism subject to periodic random denial of service (DoS) jamming attacks and infinitely distributed delays. In order to characterize the occurrence of periodic random DoS jamming attacks in the network channel between controller and actuator, the Kronecker delta function is used to represent the periodic switching between the sleeping period and attack period, and a Bernoulli-distributed random variable is utilized to reflect the probabilistic occurrence of DoS attacks. Infinitely distributed delay is involved to reflect actual state lag. The relative event-triggering mechanism is employed to reduce unnecessary information transmission and save communication energy in the network channel between sensor and observer. An observer-based PID controller is constructed for the regulation of the system to achieve an appropriate working effect. The aim of this paper is to design a security-guaranteed PID controller for delayed systems such that both the exponential mean-square stability and the H∞ performance are satisfied. Using the Lyapunov stability theory, stochastic analysis method and matrix inequality technique, a sufficient condition is put forward that ensures the existence of the required observer and PID controller. Gain parameters of the observer and the PID controller are computed by solving a certain matrix inequality. A simulation is carried out to verify the effectiveness of the developed observer-based H∞ PID control method. The obtained H∞ noise rejection level is below 0.85, the average event-based release interval is 13, the absolute values of the maximum estimation error of two elements in the system state are 1.434 and 0.371 using the observer, and two elements of the system state converge to 0.238 and −0.054 at the 41th time step with two elements of the control output being 0.031 and 0.087.

A Diffusion Model Analysis of Object-Based Selective Attention in the Eriksen Flanker Task.

Selective attention might be space-, feature-, and/or object-based. Clear support for the involvement of an object-based mechanism is rather scarce, possibly because the predictions of models from these different classes often overlap. Yet, only object-based models can account for a larger congruency effect (CE) in the Eriksen flanker task when flankers are more (vs. less) strongly grouped to the target, but spacing and other response-irrelevant features of target and flankers are held constant. Exactly this was observed by Kramer and Jacobson (1991). So far, this theoretically relevant finding has not been replicated closely. We replicated the finding in two web-based experiments. Specifically, CEs were larger when flanker lines were connected to the central target line (vs. to outer neutral lines). We also successfully fitted the Diffusion Model for Conflict tasks (DMC) to the experimental data. Critically, diffusion modeling (DMC) and distributional analyses (delta functions) revealed that object membership primarily affected target processing strength rather than strength or timing of flanker processing. This challenges the prominent attentional spreading (sensory enhancement) account of object-based selective attention and motivates an alternative target attenuation account.

Full-Waveform Inversion for Breast Ultrasound Tomography Using Line-Shape Modeled Elements.

The objective of the work described here was to incorporate the spatial shapes of the transducer elements into the framework of the full-waveform inversion. An element is treated as its cross-section in the 2-D imaging plane, that is, a line segment. The elements are not simply modeled as a set of point sources on their surface to avoid staircasing artifacts. By use of the Fourier collocation method, an element is spatially represented as the discrete convolution between its spatial distribution and a band-limited delta function. The excitation pulses on the emitters and recorded signals on the receivers are then weighted based on the discrete convolution results. Digital and physical experiments are implemented to validate the method. It is meaningful to model the shapes of the elements if their spatial sizes are similar to or larger than the acoustic wavelengths. It should, however, be noted that because this article focuses on 2-D imaging, the inter-plane effects are not considered. The approach helps reduce the root mean square errors and increase the structural similarity of the reconstructed images. It also helps to improve the stability of convergence and to accelerate the convergence speed.

Development of a Moving-Least-Square-based Immersed Boundary Method for the simulation of viscous Compressible Flows

In this paper, a Moving-Least-Squares (MLS)-based immersed boundary method (IBM) for simulating viscous compressible flows is proposed. For this method, the compressible lattice Boltzmann flux solver (LBFS) is used to solve the flow field on the Eulerian mesh and the MLS-based IBM introduced in present paper is used to implement boundary condition. The discrete Lagrangian points are used to represent the Solid boundaries. A second-order MLS interpolation is used for numerical exchange between two grids of IBM. The forced implicit boundary condition is adopted to ensure that the physical boundary condition can be accurately satisfied. The present method is able to achieve a higher numerical accuracy than traditional IBM, which use the Delta function for interpolation. Some numerical validation tests are used for the validation of proposed method. The order and accuracy of the MLS approximation are tested by using a classical example of flow past a cylinder. Test of flow past the NACA0012 airfoil at different Mach numbers, angles of attack and Reynolds numbers is applied to verified the accuracy.

Thermodynamic properties and superstatistics analysis of particle in the non-relativistic system under the influence of trigonometric scarf potential using Dong proper quantization

There has been an intense study on the eigenvalue and eigenfunction of the nonrelativistic quantum system. The Schrodinger equation is important in the nonrelativistic quantum system that is used to analyze the quantum information of the system, so it attached the interest of the researchers to conduct the exploration on it. In most of the studies, the researchers expand the study of the solution of the Schrodinger equation to the physical properties of the system. The thermodynamical property is one of the physical properties and key to analyzing the characteristic of materials. It needs more investigation since it has the potential to form a new design of the material. In this study, the Schrodinger equation with trigonometric Scarf potential was solved using the Supersymmetry WKB quantization condition scheme, where the variable of trigonometric Scarf potential was transformed into the Poschl-Teller potential. The energy spectra were obtained by comparing the constant parameters between Trigonometric Scarf potential and Poschl-Teller potential and by using Dong proper quantization condition of transformed Supersymmetry WKB for trigonometric Scarf potential. The result shows that the energy spectra are influenced by the radial quantum number and the potential width, and the energy increase by the increase of the radial quantum number and the potential width. The thermodynamic properties of the system were obtained approximately using the energy spectra equation and were expressed in the erf function. The superstatistics mechanics was approximately obtained by using modified Delta Dirac function distribution for the Boltzmann factor. These properties are important to analyze the characteristic of the designed solids.

Accurate and Efficient Simulation of Very High-Dimensional Neural Mass Models with Distributed-Delay Connectome Tensors

This paper introduces methods and a novel toolbox that efficiently integrates high-dimensional Neural Mass Models (NMMs) specified by two essential components. The first is the set of nonlinear Random Differential Equations (RDEs) of the dynamics of each neural mass. The second is the highly sparse three-dimensional Connectome Tensor (CT) that encodes the strength of the connections and the delays of information transfer along the axons of each connection. To date, simplistic assumptions prevail about delays in the CT, often assumed to be Dirac-delta functions. In reality, delays are distributed due to heterogeneous conduction velocities of the axons connecting neural masses. These distributed-delay CTs are challenging to model. Our approach implements these models by leveraging several innovations. Semi-analytical integration of RDEs is done with the Local Linearization (LL) scheme for each neural mass, ensuring dynamical fidelity to the original continuous-time nonlinear dynamic. This semi-analytic LL integration is highly computationally-efficient. In addition, a tensor representation of the CT facilitates parallel computation. It also seamlessly allows modeling distributed delays CT with any level of complexity or realism. This ease of implementation includes models with distributed-delay CTs. Consequently, our algorithm scales linearly with the number of neural masses and the number of equations they are represented with, contrasting with more traditional methods that scale quadratically at best. To illustrate the toolbox's usefulness, we simulate a single Zetterberg-Jansen and Rit (ZJR) cortical column, a single thalmo-cortical unit, and a toy example comprising 1000 interconnected ZJR columns. These simulations demonstrate the consequences of modifying the CT, especially by introducing distributed delays. The examples illustrate the complexity of explaining EEG oscillations, e.g., split alpha peaks, since they only appear for distinct neural masses. We provide an open-source Script for the toolbox.

The algebraic origin of the Doppler factor in the Liénard–Wiechert potentials

After reviewing the algebraic derivation of the Doppler factor in the Liénard–Wiechert potentials of an electrically charged point particle, we conclude that the Dirac delta function used in electrodynamics must be the one obeying the weak definition, non-zero in an infinitesimal neighborhood, and not the one obeying the strong definition, non-zero in a point. This conclusion emerges from our analysis of (a) the derivation of an important Dirac delta function identity, which generates the Doppler factor, (b) the linear superposition principle implicitly used by the Green function method, and (c) the two equivalent formulations of the Schwarzschild–Tetrode–Fokker action. As a consequence, in full agreement with our previous discussion of the geometrical origin of the Doppler factor, we conclude that the electromagnetic interaction takes place not between points in Minkowski space, but between corresponding infinitesimal segments along the worldlines of the particles.

Microlocal analysis for Gelfand–Shilov spaces

We introduce an anisotropic global wave front set of Gelfand–Shilov ultradistributions with different indices for regularity and decay at infinity. The concept is defined by the lack of super-exponential decay along power type curves in the phase space of the short-time Fourier transform. This wave front set captures the phase space behaviour of oscillations of power monomial type, a k a chirp signals. A microlocal result is proved with respect to pseudodifferential operators with symbol classes that give rise to continuous operators on Gelfand–Shilov spaces. We determine the wave front set of certain series of derivatives of the Dirac delta, and exponential functions.

Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schrödinger framework to solve the Liouville equation for the HJE with the Hamiltonian formulation of classical mechanics, since it can be recast as the semiclassical limit of the Wigner transform of the Schrödinger equation. Comparison between the Schrödinger and the Liouville framework will also be made.

Missing corner in the sky: massless three-point celestial amplitudes

We present the first computation of three-point celestial amplitudes in Minkowski space of massless scalars, photons, gluons, and gravitons. Such amplitudes were previously considered to be zero in the literature because the corresponding scattering amplitudes in the plane wave basis vanish for generic momenta due to momentum conservation. However, the delta function for the momentum conservation has support in the soft and colinear regions, and contributes to the Mellin and shadow integrals that give non-zero celestial amplitudes. We further show that when expanding in the (shadow) conformal basis for the incoming (outgoing) particle wave functions, the amplitudes take the standard form of correlators in two-dimensional conformal field theory. In particular, the three-point celestial gluon amplitudes take the form of a three-point function of a spin-one current with two spin-one primary operators, which strongly supports the relation between soft spinning particles and conserved currents. Moreover, the three-point celestial amplitudes of one graviton and two massless scalars take the form of a correlation function involving a primary operator of conformal weight one and spin two, whose level-one descendent is the supertranslation current.

On Classical and Distributional Solutions of a Higher Order Singular Linear Differential Equation in the Space K’

In this research work, we aim to find and describe all the classical solutions of the homogeneous linear singular differential equation of order l in the space of K' distributions. Recall that in our previous research, the results of which have been published in some journals, we had undertaken similar studies in the case of a singular differential equation of the Euler type of second order, when the conditions were carried out. That said, our intentions in this article are therefore to generalize the results obtained and recently published, focusing our research on the situation of the homogeneous singular linear differential equation of order l of Euler type. In this orientation, we base ourselves on the classical theory of ordinary linear differential equations and look for the particular solution to the equation considered in the form of the distribution with a parameter to be determined, which we replace in the latter. Depending on the nature of the roots of the characteristic polynomial of the homogeneous equation we identify, case by case, all the solutions indicated in the sense of distributions in the space K'. In this same work, we return to the non-homogeneous equation of order l of the same Euler type, whose second member consists only of the derivative of order s of the Dirac-delta distribution studied in our previous work, to fully describe all the solutions of the latter in the sense of distributions in the space K'. We finalize this work by making an important remark emphasizing the interest in undertaking research of the same objective of finding a general solution, by studying the singular differential equations of the same higher-order l with the particularity of being of Euler types on the left and Euler on the right in the space of distributions K’.