Abstract Picard's method of successive approximations is utilized to address the linear differential equation that governs the behaviour of the damped forced harmonic oscillator under the influence of an arbitrary forcing function. With the time-dependent substitution $\chi(t) = e^{-\gamma t}x(t)$, the first derivative term vanishes and the differential equation reduces to a second-order ODE without damping for $x(t)$. This modified equation takes the form of an inhomogeneous Volterra integral equation of the second type, incorporating two initial conditions. Initiating the iterative process of successive approximations with a meticulously selected zero-order approximation, we demonstrate the emergence of convergent sequences of partial sums, culminating in the derivation of a closed-form solution. The straightforward approach employed in this paper to analyze the damped and forced harmonic oscillator is accessible to undergraduate students in physics, engineering, and mathematics, without requiring sophisticated mathematical tools. Moreover, it facilitates a direct physical analysis of the problem and serves as an insightful example of the application of Taylor series expansion. As specific examples, presented merely as illustrative applications that have not yet been addressed in the literature (at least not in this context for the second example), we consider an oscillator driven by a Gaussian-like force and by a normalized exponential force. These examples can be seen as regularized generalizations of an impulsive, delta-like force.

We show that it is impossible to quantify the decay rate of a semi-uniformly stable operator semigroup based on sole knowledge of the spectrum of its infinitesimal generator. More precisely, given an arbitrary positive function r r vanishing at ∞ \infty , we construct a Banach space X X and a bounded semigroup ( T ( t ) ) t ≥ 0 (T(t))_{t \geq 0} of operators on it whose infinitesimal generator A A has empty spectrum σ ( A ) = ∅ \sigma (A)=\varnothing , but for which, for some x ∈ X x \in X , lim sup t → ∞ ‖ T ( t ) A − 1 x ‖ X r ( t ) = ∞ . \begin{equation*} \limsup _{t\to \infty } \frac {\|T(t)A^{-1}x\|_{X}}{r(t)}=\infty . \end{equation*}

This paper proposes a generalized optimal impact angle guidance law against axially maneuvering targets. The guidance problem is formulated as a finite-time optimal control problem based on collision triangle geometry, and an analytical solution is derived. By introducing an arbitrary positive weighting function, the guidance command can be flexibly shaped to meet various mission requirements. The convergence of the guidance law is rigorously proven via Lyapunov stability theory. Extensive simulations, including various engagement geometries, complex composite axial target maneuvers, and parameter uncertainties, as well as comparative studies against existing advanced guidance laws, are conducted. The results comprehensively demonstrate the accuracy, robustness, and flexibility of the proposed method for intercepting axially maneuvering targets.

We demonstrate the time evolution of a free particle in a three-dimensional space given that the initial condition is any arbitrary radial function f(r). The solution is expressed as a series expansion in terms of generalized Bessel functions derived using a 3D recursion formula for Bessel functions in the radial coordinate r. Additionally, we establish that these generalized Bessel functions can be represented through intricate double series, which ultimately enable the construction of the full solution. This work presents a novel solution to the problem, as previous approaches were limited to expressions involving only spherical Bessel functions.

The Lemaître–Tolman–Bondi (LTB) solution to the Einstein equations describes the dynamics of a self-gravitating spherically symmetric dust cloud with an arbitrary density profile and any distribution of initial velocities, encoded in three arbitrary functions f(R), F(R), and τ0(R), where R is a radial coordinate in the comoving reference frame. A particular choice of these functions corresponds to a wormhole geometry with a throat defined as a sphere of minimum radius at a fixed time instant. In this paper we explore LTB wormholes and discuss their possible observable appearance, studying in detail the effects of gravitational lensing by such objects. For this aim, we study photon motion in wormhole space-time inscribed in a closed Friedmann dust-filled universe and find the wormhole shadow as it could be seen by a distant observer. Because the LTB wormhole is a dynamic object, we analyze the dependence of its shadow size on the observation time and on the initial size of the wormhole region. We reveal that the angular size of the shadow exhibits a non-monotonic dependence on the observation time. At early times, the shadow size decreases as photons with smaller angular momentum gradually reach the observer. At later times, the expansion of the Friedmann universe becomes a dominant factor that leads to an increase in the shadow size.

Quantizing nanolaminates are under development for use as materials with optimized properties for optical interference coatings. The optical dispersion needs a different description compared with simply mixing the basic materials because of the band gap shift. Although quantizing nanolaminates consist of alternating high- and low-index materials, representing them as a conventional layer stack can be questionable due to their extremely thin—often nanometer-scale—layers. This work gives an overview of Si-SiO 2 layer stacks over a large range of different thickness combinations. The samples range from an almost pure SiO 2 single layer to an almost standard interference stack of both materials. By treating all samples as single layers and fitting their dispersions to an arbitrary function, we examined which combinations show a shift in the absorption edge and can be described with sufficient accuracy that they are suitable for production.

This paper studies a mixed PDE containing the second time derivative and a quadratic nonlinearity of the Monge–Ampère type in two spatial variables, which is encountered in geophysical fluid dynamics. The Lie group symmetry analysis of this highly nonlinear PDE is performed for the first time. An invariant point transformation is found that depends on fourteen arbitrary constants and preserves the form of the equation under consideration. One-dimensional symmetry reductions leading to self-similar and some other invariant solutions that described by single ODEs are considered. Using the methods of generalized and functional separation of variables, as well as the principle of structural analogy of solutions, a large number of new non-invariant closed-form solutions are obtained. In general, the extensive list of all exact solutions found includes more than thirty solutions that are expressed in terms of elementary functions. Most of the obtained solutions contain a number of arbitrary constants, and several solutions additionally include two arbitrary functions. Two-dimensional reductions are considered that reduce the original PDE in three independent variables to a single simpler PDE in two independent variables (including linear wave equations, the Laplace equation, the Tricomi equation, and the Guderley equation) or to a system of such PDEs. A number of specific examples demonstrate that the type of the mixed, highly nonlinear PDE under consideration, depending on the choice of its specific solutions, can be either hyperbolic or elliptic. To analyze the equation and construct exact solutions and reductions, in addition to Cartesian coordinates, polar, generalized polar, and special Lorentz coordinates are also used. In conclusion, possible promising directions for further research of the highly nonlinear PDE under consideration and related PDEs are formulated. It should be noted that the described symmetries, transformations, reductions, and solutions can be utilized to determine the error and estimate the limits of applicability of numerical and approximate analytical methods for solving complex problems of mathematical physics with highly nonlinear PDEs.

Hermite polynomials facilitating on-line learning analysis of layered neural networks with arbitrary activation function

We study an integral transform—here called the Master Integral Transform—in which the kernel is an arbitrary entire function of finite order. When the nonzero Taylor coefficients of the kernel have positive Beurling–Malliavin density, we prove completeness and global injectivity in a Cauchy-weighted Hilbert space, and we furnish explicit Mellin–Fourier inversion formulae with exponentially decaying integrands. Classical Fourier, Laplace, and Mellin transforms appear only as strict special cases. Beyond these, we establish structural properties (multiplier/composition law, dilation covariance, parameter regularity) and present applications not captured by fixed-kernel frameworks, including inverse-kernel identification and hybrid boundary value models, e.g., the Poisson–Airy pair produces a closed-form transformed Green’s function and a solvable variable-coefficient PDE, illustrating capabilities unavailable to fixed-kernel frameworks.

This study investigates the thermoelastic frictional contact and wear behavior during reciprocating sliding of a conductive cylindrical punch on a functionally graded piezoelectric material (FGPM)-coated half-plane. The thermo-electro-elastic properties of the coating vary continuously along the thickness direction according to arbitrary gradient functions, with thermal parameters being temperature-dependence. A theoretical framework for the coupled thermo-electro-elastic frictional contact problem is developed and solved using the finite element method. A sequential coupling approach is employed to integrate thermoelastic frictional contact with piezoelectric effects. Furthermore, wear on the coating surface is modeled using an improved Archard formulation, accounting for its impact on thermal sliding frictional contact characteristics. Numerical simulations examine the influence of wear, cycle number, friction coefficient, gradient index and gradient form on the coupled thermo-electro-elastic response of the FGPM coating structure. The numerical results demonstrate the gradient index and gradient form can effectively mitigate thermo-electrical contact-induced damage and reduce friction and wear in piezoelectric materials.

Dispersion, nonlinearity, and waves-wave interactions are some of the important aspects of two-dimensional wave behaviour that are captured by the mathematical model known as the ion sound and Langmuir wave system. This study uses the truncated Painlevé technique to examine the (1+1) dimensional integrable ion Sound and Langmuir wave system. Various localised solutions, such as rogue or irregular waves, dromion-pair, and dromions can be produced via applying random functions to the findings. The collisional interaction of these solutions are produced, investigated, and visually displayed as a result of selecting suitable starting values for the arbitrary functions. We discover that dromions interact inelastically, exchanging both energy and phase, but rogue waves are inherently erratic. These findings advance our understanding of complex wave dynamics and have important implications for the study of non-linear occurrences in a variety of disciplines such as physical mechanisms, fluid dynamics, oceanography and nonlinear optics. It’s crucial to remember that all computations and visualisations are generated and their reliability and accuracy verified using Mathematica software. All things taken into account this work enhances our knowledge of complexities nonlinear systems and how their behaviour is influenced by their initial conditions. The study’s findings will aid in our comprehension of how waves behave in higher dimensional controlling models.

In Frozen Density Embedding Theory (FDET) [T. A. Wesolowski, Phys. Rev. A 77, 012504 (2008)], the total N-electron density is represented as a sum of two components ρ1 and ρ2, where ρ1 is obtained from a Schrödinger-like N'-electron eigenvalue equation with N' < N and ρ2 is an arbitrary non-negative real function integrating with respect to N - N'. It is shown that the exact total ground-state electron density ρvo cannot be obtained from FDET even if ρ2 is piecewise equal to ρvo (i.e., equal to ρvo on some measurable volume element). The result is discussed in the context of subsystem approach in density functional theory, pseudopotential theory, and embedding potentials derived from inverted Kohn-Sham equation.

In this paper, we give a decomposition of the gradient measure [Formula: see text] of an arbitrary function of bounded variation [Formula: see text] into a linear combination of atoms [Formula: see text], where [Formula: see text] is a set of finite perimeter. The atoms further satisfy the support, cancellation, normalization, and size conditions: For each [Formula: see text], there exists a cube [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text], and, denoting by [Formula: see text] the heat kernel in [Formula: see text], [Formula: see text] Our proof relies on a sampling of the coarea formula and a new boxing identity. We present several consequences of this result, including Sobolev inequalities, dimension estimates, and trace inequalities.

Abstract Precise and compact polarization control is vital in fields ranging from classical optics to high‐dimensional physics. Fiber‐based polarization optics offer low‐loss and high flexibility but are limited in miniaturization and functional diversity, while planar integrated optics often suffer from polarization sensitivity and poor fiber compatibility. A 3D polarization optics platform is presented, based on laser‐written gradient‐index (GRIN) waveguides in bulk glass. This approach enables high‐fidelity polarization engineering with record‐low birefringence (2 × 10 −8 )—over an order of magnitude lower than standard fibers—and low coupling losses (as low as 0.02 dB per facet). Besides, this platform supports the high‐fidelity implementation of polarization‐encoded 3D photonic circuits, integrating essential polarization operations such as splitting, conversion, and arbitrary waveplate functions, by customizing polarization‐dependent GRIN waveguides with birefringence accuracy of 10 −8 and arbitrary rotation of the optical axis over a full 0‐π range. Leveraging these capabilities, controlled polarization retardance covering the entire fiber‐optic communication band (1030–1700 nm) is further demonstrated, and realize a unidirectional waveguide array via spatially engineered birefringence and coupling strength, achieving &gt;20 dB crosstalk suppression without external fields. This platform bridges the gap between fiber and chip‐scale photonics, offering a versatile route to broadband optical interconnects, polarization‐encoded quantum optics, and nonreciprocal topological photonics.

This study examines how the Laplace transform changes when its kernel base is expressed as an arbitrary exponential function. The results show its effectiveness in processing the Dirac-delta function.

The article is devoted to the construction of fast neurons and neural networks for the implementation of two complete logical bases and modeling of computing devices on their basis. The main idea is to form a fast activation function based on semi-parabolas and its variations that have effective computational support. The constructed activation functions meet the basic requirements that allow configuring logical circuits using the backpropagation method. The main result is obtaining complete logical bases that open the way to constructing arbitrary logical functions. Models of such elements as a trigger, a half adder, and an adder, which form the basis of various specific computing devices, are presented and tested. It is shown that the new activation functions allow obtaining fast solutions with a slight decrease in quality compared to reference outputs. To standardize the outputs, it is proposed to combine the constructed circuits with a unit jump activation function.

Abstract For any given set , we discuss a fractal frequency‐localized version of the local smoothing estimates for the half‐wave propagator with times in . A conjecture is formulated in terms of a quantity involving the Assouad spectrum of and the Legendre transform. We validate the conjecture for radial functions. We also prove a similar result for fractal‐time and square function bounds, for arbitrary functions and general time sets. We formulate a conjecture for generalizations.

ABSTRACT The surface motion during earthquakes can undergo significant changes owing to the two-dimensional field effect. In addition to the terrain effect, the nonuniformity of the surrounding soil density can significantly affect the dynamic response of a site to earthquakes. In this study, the wave function expansion method and Graf’s addition formula are used to derive an analytical solution for the dynamic response of a cutting site under SH wave incidence with the soil density represented by an arbitrary function. The displacement amplitude results for the cutting site surface under a radial density distribution are subsequently presented. The seismic responses of isosceles-shaped road cuts are investigated under various incident angles, dimensionless frequencies, cutting slope, and density parameters. The results indicate that the nonuniformity of the soil density around road cuts can significantly alter the dynamic responses of local site surfaces under seismic action. One of the most important results is that, compared with the case of soil with a homogeneous density, the nonuniformity of the soil density around the soil cutting does not independently influence the various parameters of the seismic response results. The presence of density parameters exacerbates the seismic interaction between the soil and the structure. In addition, the results of simplifying the model to a flat ground model indicate that the nonlinear effect of the soil causes amplification of shear waves during propagation.

This paper considers one-dimensional equations of acoustics equations of inhomogeneous media and the system of gas dynamics equations with constant entropy. Using the Riemann approach, the gas dynamics equations are reduced to a second-order linear hyperbolic equation with variable coefficients. Solutions to this equation are constructed using Euler–Darboux transformations. This allows us to find new exact solutions of the equations of acoustics and gas dynamics, depending on two arbitrary functions.

The aim of the study is to further develop analytical methods for the bending analysis of beams resting on an inhomogeneous (variable) continuous Winkler elastic foundation. The inhomogeneity of the foundation is characterized by a spatially varying bedding (subgrade reaction) modulus. This work considers a generalized case in which the flexural rigidity, foundation modulus, and external loading are defined as arbitrary continuous functions of the coordinate along the beam’s centerline. Exact expressions are derived for the fundamental functions and a particular solution of the corresponding fourth-order differential equation with variable coefficients. These functions are dimensionless and are represented as absolutely and uniformly convergent power series in a dimensionless parameter, with variable coefficients determined using recurrent integral relations. The stress–strain state (SSS) parameters of the beam–deflection, rotation angle, bending moment, and shear force–are expressed through the aforementioned functions. The unknown integration constants in these expressions are determined from the prescribed boundary conditions. For practical application, both the fundamental functions and the particular solution are transformed into a power series format. As a result, the bending analysis of the beam reduces to a numerical implementation of explicit analytical formulas for the SSS parameters. A software implementation of the derived formulas was developed using Visual Basic within the Excel environment, thus enabling beam analysis in a computational mode. A practical example is provided to demonstrate the application of the obtained solutions. Calculations are performed for a beam with free ends shaped as a truncated pyramid, whose width and height vary linearly. The results are presented both numerically and graphically. The obtained numerical values are exact, as the proposed approach is based on an exact solution to the corresponding differential equation. The availability of such solutions allows for the assessment of the accuracy of approximate methods by direct comparison.