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The study is devoted to some applied aspects of the Dirac delta function. On the basis of this function, an integral representation was found for the deviation of the functions of the Holder class H α (0 < α < 1) from their Poisson integrals in the upper half-plane. In the current research, exact equalities of the upper bounds for the deviations of the functions of the Holder class H α from the Poisson operators in the upper half-plane were found by applying the known properties of the Dirac delta function.

Noise emitted by rolling tires has a strong contribution on the traffic noise in urban areas and has a significant impact on the ride comfort for passengers, which is mainly caused by the interaction between tires and pavement. The force generated in the contact patch has an important contribution to tire noise. For a patterned tire rolling on a flat surface, the geometry and position of tread pattern blocks are one of the main influencing factors of discontinuous force. In this paper, it is assumed that the impact force produced by each tread pattern element and pavement can be expressed as a Dirac delta function. For more realistic modelling of noise generated by impact force, a weighting function is considered, which is distributed along the tread transverse direction. The experiment is performed on a drum experimental machine in a semi-anechoic room. The predicted results are compared with experimental measurement, the maximum error is 3.85 dB between the predicted sound pressure level and measured. The prediction results have sufficient accuracy on the trend of A-weighted sound pressure level. The prediction method in this paper can be used to accelerate the development process of low tire tread pattern in the future.

Many problems involving internal interfaces can be formulated as partial differential equations with singular source terms. Numerical approximation to such problems on a regular grid necessitates suitable regularizations of delta functions. We study the convergence properties of such discretizations for constant coefficient elliptic problems using the immersed boundary method as an example. We show how the order of the differential operator, order of the finite difference discretization, and properties of the discrete delta function all influence the local convergence behavior. In particular, we show how a recently introduced property of discrete delta functions—the smoothing order—is important in the determination of local convergence rates.

We perform a comparative analysis of different computational approaches employed to explore the electronic structure of ultralong-range Rydberg molecules. Employing the Fermi pseudopotential approach, where the interaction is approximated by an s-wave bare delta function potential, one encounters a non-convergent behavior in basis set diagonalization. Nevertheless, the energy shifts within the first order perturbation theory coincide with those obtained by an alternative approach relying on Green's function calculation with the quantum defect theory. A pseudopotential that yields exactly the results obtained with the quantum defect theory, i.e. beyond first order perturbation theory, is the regularized delta function potential. The origin of the discrepancies between the different approaches is analytically explained.

Wavefields that are related to a source point with complex coordinates, which are not all real numbers, meanwhile have a long history. They have first been applied in electromagnetic theory, and subsequently also in acoustics. For example, such wavefields exhibit a directivity whose degree can be controlled by the imaginary part of the source point. The wavefields in question are obtained if in the fundamental solution for a real source point this point is replaced by its complex counterpart. With respect to the complex source point, sometimes it is mystically spoken about the associated complex delta function. In a rigorous way, however, only an equivalent source, which is defined on the real space, could be derived. In the present paper, we show that a wavefield, which is related to a complex source point and solves the Helmholtz equation, can be extended to an ultradistributional solution of the Helmholtz equation with the complex delta function as right-hand side. This result demonstrates that the connection between those wavefields and the complex delta function is not only a formal one. Our analysis is performed in one, two and three space dimensions.

The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function in the presence of the drift term. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains and discuss how the introduction of the transport term influences the regularity of the solutions.

The propagation of waves in a homogeneous, isotropic linear elastic half-space is investigated herein. A buried point pulse generates a vibration in an arbitrary direction. Both source and receiver points are located in the interior of the three-dimensional domain. To solve the elastodynamic problem, also known as Lamb's problem, the source image and the superposition principle are both used to derive a numerical transient solution. Accordingly, the transient response of the problem in the time domain can be considered as the superposition of the responses to the real and imaginary sources in the full-space and some additional vertical loads on the surface of the half-space. The additional vertical loads are distributed over a rectangular area on the surface of the half-space and are space- and time-dependent functions that vary with time as the Heaviside step, the Dirac delta, and derivatives of the Dirac delta functions. The motion at depth due to a point source applied on the surface is obtained using some well-known approaches reported in the literature. To achieve the Laplace transform displacement, Helmholtz potentials have been employed for the displacement field and the Laplace transform wave equation as well as the Hankel transform of boundary conditions have been satisfied. The inverse Laplace transform (time-domain solutions) is found via a modified version of some other methods reported in the literature. The solutions obtained in this way automatically satisfy the traction-free boundary conditions over the surface of the half-space and can be implemented in the three-dimensional time-domain boundary element method (BEM), and no discretization of the ground surface is needed.

A novel approach to constructing an algorithm for computing discrete logarithms, which holds significant interest for advancing cryptographic methods and the applied use of multivalued logic, is proposed. The method is based on the algebraic delta function, which allows the computation of a discrete logarithm to be reduced to the decomposition of known periodic functions into Fourier–Galois series. The concept of the “partial discrete logarithm”, grounded in the existence of a relationship between Galois fields and their complementary finite algebraic rings, is introduced. It is demonstrated that the use of partial discrete logarithms significantly reduces the number of operations required to compute the discrete logarithm of a given element in a Galois field. Illustrative examples are provided to demonstrate the advantages of the proposed approach. Potential practical applications are discussed, particularly for enhancing methods for low-altitude diagnostics of agricultural objects, utilizing groups of unmanned aerial vehicles, and radio geolocation techniques.

The aim of this paper is to establish an Ambrosetti–Prodi type result involving Dirac weights u ′ ′ ′ ′ ( x ) + q ( x ) u ( x ) = ( c ( x ) + ∑ i = 1 p c i δ ( x - x i ) ) ( g ( u ( x ) ) + f ( x ) ) , x ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = u ′ ′ ( 0 ) = u ′ ′ ( 1 ) = 0 , where δ ( x - x i ) is the canonical Dirac delta function at the point x i , i = 1 , 2 , … , p , p ∈ N , 0 = x 0 < x 1 < ⋯ < x p < x p + 1 = 1 , q ∈ C ( [ 0 , 1 ] , [ 0 , + ∞ ) ) , f ∈ L 1 ( [ 0 , 1 ] , R ) , g ∈ C 1 ( R , R ) , c ∈ C ( [ 0 , 1 ] , [ 0 , + ∞ ) ) , c i ∈ [ 0 , + ∞ ) . The main tools used are the sub-super-solution method and Leray–Schauder topological degree theory.

Kelvin–Voigt equations for inhomogeneous fluids with a singular right side are studied. A singular term that approximates the Dirac delta function on an initially infinitely thin layer is introduced into the right side of a mass balance equation. This singular term is similar to the relaxation term used to describe nonequilibrium processes in hydrodynamics. In an extreme case, when a small parameter, namely, the characteristic size of the initial layer, tends to zero, the density and velocity at the initial time change abruptly.