Search Results (340)

This paper investigates a Sturm–Liouville problem formulated in terms of the diamond-$\alpha$ derivative on a uniform time scale. The proposed setting combines the delta and nabla components within a unified spectral framework. A key feature of this framework is the explicit identification of the defect structure arising from diamond-$\alpha$ integration by parts, which distinguishes the resulting spectral theory from its classical continuous counterpart. Under appropriate regularity and uniqueness assumptions, a conditional linear-dependence criterion is established for solutions corresponding to the same eigenvalue, whereas eigenfunctions associated with distinct eigenvalues satisfy an orthogonality relation under a suitable compatibility condition. A generalized Green identity is derived, and the resulting remainder term is shown to define a skew-symmetric defect form. Moreover, an explicit estimate for this term is obtained, yielding $R_{\alpha }(u,v)=O\left(\nu \right)$ as the graininess $\nu$ tends to zero under uniform boundedness assumptions. When the boundary form vanishes and the spectral gap remains uniformly separated from zero, this estimate further implies asymptotic orthogonality of eigenfunctions associated with distinct eigenvalues in the dense-limit regime. A generalized integral identity for the eigenfunctions is also established. The classical continuous structure is recovered when $\mathbb{T}=\mathbb{R}$, while the defect term vanishes in the dense-limit regime under the stated boundedness assumptions. Full article

We quantize the exterior region of an odd-dimensional Kerr-AdS black hole, in which all rotational parameters are equal, using our model of quantum gravity. The resulting hyperbolic equation is solved by products of temporal eigenfunctions $w_i$, whose corresponding eigenvalues all have multiplicity one and spatial eigendistributions $v_{ij}$ corresponding to the same eigenvalues but having multiplicities $m_i\ge 1$. In principle, the $m_i$ may be arbitrarily large. When only the exterior region is considered, there is no criterion for determining the values of the $m_i$. However, when the quantization of the interior region is also considered, the same ambiguity does not arise, because the multiplicities can be chosen to be maximal. It is therefore natural to use the same values in the exterior region. Because the temporal Hamiltonian is the same in both regions, the interior and exterior eigenvalues coincide, and this choice defines a unitary equivalence between the respective Hilbert spaces and Hamiltonians. The unitary equivalence between the Hilbert spaces and their respective Hamiltonians preserves the complete quantum-statistical information carried by all normal states in the interior and exterior regions of the black hole. Hence, there is no information paradox at the quantum level. Full article

Public health outcomes can be heavily influenced by the landscape of public opinion; hence, it is important to understand how that landscape changes over time. For one, opinions on public health issues are responsive to official pronouncements, whether from the governmental or professional medical establishments. Additionally, in today’s world of high speed communication, opinion can also be highly responsive to the broadcast opinions of “influencers” whose large numbers of followers assure them of a broad reach. To understand the opinion landscape in a general sense, we develop an ordinary differential equation model for opinion change that is based primarily on attraction to prominent sources whose opinions are independent of the opinions of others. The individual opinion change model is then used to develop a Fokker–Planck-type partial differential equation model for the overall opinion landscape. This model is shown to have a stable equilibrium solution, and the dependence of the equilibrium solution on key model parameters is illustrated with examples based on opinion regarding vaccination. Eigenvalues and eigenfunctions for the associated Sturm–Liouville problem are determined numerically by two methods, one by using numerical solutions of the partial differential equation and the other by applying a shooting method directly to the eigenvalue problem. Full article

In the present work, we investigate a $\beta$-Dirac problem on the whole-line. Our primary objectives are threefold: to demonstrate the existence of a spectral function, to deduce the associated Parseval identity, and to formulate an eigenfunction expansion theorem that characterizes the behavior of the problem under consideration. Full article

To address the limitations of traditional analytical models in characterising multi-core coupling effects and the excessive computational cost of finite element simulations, this paper presents a high-precision analytical model for a novel T-II composite-core eddy current testing (ECT) probe using the Truncated Region Eigenfunction Expansion (TREE) method. The analytical expressions of coil impedance are derived by partitioning the solution domain into ten subdomains under an axisymmetric cylindrical coordinate system, with rigorous satisfaction of electromagnetic continuity at all material interfaces. Numerical cross-validation against 2D and 3D Finite Element Method (FEM) simulations under idealised modelling assumptions across the frequency range of 100 Hz to 10 kHz shows that the proposed TREE model yields relative errors below 2% for both coil resistance and reactance. Notably, the proposed approach requires significantly less computation time than 2D and 3D FEM. Further parametric analysis confirms that the proposed T-II composite-core probe delivers superior electromagnetic performance compared to conventional single-core probes, including intensified subsurface eddy current densities and improved magnetic field redistribution. This work overcomes the inherent limitations of single-core ECT analytical models, establishes a robust theoretical paradigm to interpret the distinctive electromagnetic field advantages of composite-core probes, and provides solid support for the structural optimisation of multi-core ECT sensors. Full article

This paper investigates an inverse problem of recovering the spatial component of a source term for a degenerate ultraslow diffusion equation with a Hadamard–Caputo fractional derivative. Nonlocal boundary conditions of the Samarskii–Ionkin type are imposed. A distinctive feature of the problem is that the system of eigenfunctions of the associated spectral problem, although complete and minimal, does not form an unconditional basis in $L_2(0,1)$. Hence the standard eigenfunction Fourier expansion cannot be applied directly. To overcome this difficulty, we construct a special auxiliary system from normalized linear combinations of asymptotically close eigenfunctions and prove that it forms a Riesz basis. Expansion with respect to this basis reduces the inverse problem to a cascade system of fractional differential equations for the modal coefficients. Using spectral asymptotics, estimates for the Le Roy function, and two-sided Riesz basis inequalities, we prove the existence, uniqueness, and conditional Lipschitz stability of a strong generalized solution and rigorously justify the convergence of the resulting series. We also quantify the high-frequency instability of the inverse mapping in the weaker $L_2$ data topology. An analytical two-mode illustration shows explicitly how neglecting the coupling between asymptotically close spectral modes changes the reconstructed source coefficient. Full article

Exceptional points are parameter values at which two eigenvalues and their corresponding eigenfunctions coalesce, rendering the wave operator defective. They arise widely in non-Hermitian wave physics and disrupt the modal expansions on which semi-analytic scattering methods rely. For lined acoustic waveguides, an augmented mode-matching ansatz that restores completeness at such a degeneracy—by adjoining the generalised eigenfunction obtained from the derivative of the parametrised duct mode with respect to its transverse spectral parameter—has been established for junctions between duct sections with discrete modal sets. This article extends that ansatz to an open, radiating configuration: a rigid feed duct communicates through an impedance-lined throat, tuned to an order-two exceptional point, with a half-space bounded by a rigid flange. The radiating mouth replaces the discrete modal closure by a continuous spectrum, so the augmented basis must be matched against an improper spectral integral. The half-space field is generated by the aperture velocity, which builds the rigid-flange condition into the representation exactly, and the resulting improper integrals are rendered analytic by branch-aware substitutions whose cutoff is tied to the retained modal content. The formulation is validated on the matching and boundary conditions themselves: pointwise continuity of pressure and of normal velocity at the internal junction, pointwise pressure continuity at the radiating mouth, the vanishing of the normal velocity on the rigid flange, and the recovery of the classical flanged-duct radiation problem in the rigid limit, cross-checked against an independent implementation. The full lined problem, including the defective case, has been further verified against an independent finite-volume solution of the same boundary-value problem, whose grid-converged fractions agree with the mode-matching values to within $8\times {10}^{-5}$. A conserved-power identity is monitored as a necessary but not sufficient check. Numerical experiments confirm the known breakdown of the standard expansion at the exceptional point and the well-conditioned convergence of the augmented one in this radiating setting, and a scan of the complex admittance plane, refined by local optimisation and repeated across throat lengths and frequencies, shows that flange radiation detunes the absorption optimum away from the exceptional point, by an amount that grows with the radiated share of the power budget and vanishes as the throat lengthens. Full article

“Polymer quantum mechanics” is the name given to a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional “Schrödinger” representation. In this paper, we summarize the main features of polymer quantum mechanics and investigate in detail the polymer quantization of systems with configuration spaces that are classically compact. We show explicitly how using the standard construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and a particle in a box defined on such lattices, and discuss similarities and differences from standard Schrödinger quantum mechanics. We also explore the continuum limit of states in these systems, and demonstrate in detail how the exact eigenfunctions in the position representation approach their continuum counterparts. Full article

This investigation is concerned with a time-fractional diffusion equation governed by the Hilfer fractional derivative of order $1<{\rho }_1<2$, subject to homogeneous Neumann boundary data and generalized initial conditions of the Hilfer type. This paper addresses three interconnected problems: a direct problem and two inverse source problems (ISPs). In the first ISP, the objective is to recover an unknown space-dependent source function from measurements taken at a specified final time. In the second ISP, the goal is to determine an unknown time-dependent coefficient through an integral-type over-specification condition. By employing eigenfunction expansions in conjunction with the LT technique, we derive explicit series representations of the solutions in terms of the Mittag-Leffler function. Rigorous existence and uniqueness results for classical solutions are established for all three problems. The second ISP is reformulated as a Volterra integral equation, whose unique solvability is demonstrated via the Banach fixed point theorem. Both ISPs are shown to be ill-posed in the Hadamard sense, indicating instability with respect to data perturbations. Numerical experiments are also presented to validate the theoretical findings and to illustrate the performance of the proposed reconstruction methods. As limiting cases, the formulations corresponding to the Riemann–Liouville and Caputo fractional derivatives are recovered, illustrating the generality of the proposed framework. Full article

This study constructs a buckling analysis model integrating Euler–Bernoulli beam theory and Hamiltonian formulation to clarify the buckling characteristics of graphene origami (GOri)-reinforced beams and systematically explore the structural stability of graded composite beams. Under the symplectic space framework, the thermal buckling issue of GOri composite beams is converted into a zero-eigenvalue problem, where critical thermal buckling loads and corresponding buckling modes correspond to the symplectic eigenvalues and eigenfunctions of the Hamiltonian system. Taking the through-thickness continuity of GOri fillers into consideration, analytical expressions of buckling modes and critical buckling loads are derived using bifurcation criteria and normalization operations. Afterwards, parametric investigations are conducted to reveal how GOri content, spatial distribution, ambient temperature and folding degree affect beam buckling responses. Numerical results demonstrate that GOri distribution exerts a dominant influence on the structural buckling performance; critical thermal buckling loads tend to decline with rising folding degree and temperature. Reasonable optimization of GOri layout can significantly strengthen the mechanical capacity of composite beams, which lays solid theoretical guidance for their structural design and mechanical property enhancement. Full article

In this study, the Boscovich curve of 1763 is reinterpreted as a mean-field potential for interacting particles in condensed matter. In a dense many-body system, each particle experiences an effective potential arising from the average distribution of all the others. This mean-field potential, which exhibits alternating maxima (energy barriers) and minima (coordination shells), thereby reducing the complexity of the N-body problem to an effective two-body radial problem, with the correlation distance r as the key variable. The relationship between the PMF and the radial distribution function g(r) is given by the Kirkwood equation U_B(r) = −kT ln g(r), which provides a multi-well potential in condensed matter. Furthermore, the system is described by the Fisher density functional equation for the correlation amplitudes, −2kT ∇²ψ(r) + U_B(r)ψ(r) = μψ(r) whose eigenvalues μ_i correspond to potential levels and whose eigenfunctions ψ_i are the correlation amplitudes of the coordination shell structure. Based on the multi-well potential picture, the oscillatory behavior of U_B(r) is modeled analytically by a weighted sum of Lennard-Jones potentials, modulated by sigmoid functions. The parameters—well depths, widths, and coordination distances—are assigned on the basis of known structural properties of the system, derived either from experimental data or from geometric models such as FCC or HCP lattices. The radial distribution function is then reconstructed as a linear combination of the squared eigenfunctions obtained from the Fisher equation. The resulting discrete eigenvalue spectrum provides a spectral interpretation of the shell structure of condensed matter, wherein the complexity of many-body interactions is encoded in a hierarchy of correlation modes, each associated with a specific coordination shell. Unlike classical DFT—which relies on approximate excess free-energy functionals—and Ornstein–Zernike theory—which requires closure approximations—our approach provides a direct spectral interpretation of the coordination shell structure through the eigenvalue spectrum of the Fisher equation, where the PMF acts as the effective potential and the radial distribution function is reconstructed as a combination of squared eigenfunctions. The method is validated for liquid argon and FCC lattices and establishes a historical connection with Boscovich’s curve as a statistical potential. Full article

Drug release from long-acting intravitreal implants is governed by the coupled effects of diffusion, hydrolysis-driven erosion, and progressive shrinkage of the polymeric matrix. To capture these mechanisms, we solve the diffusion equation in a cylindrical domain whose radius decreases according to the hydrolytic degradation kinetics of PLGA, which follow a pseudo-first-order behaviour in aqueous excess. The resulting formulation combines a modal Bessel expansion with an erosion-controlled time transformation, allowing the evolving geometry and the attenuation of the diffusion modes to be incorporated in a fully mechanistic manner. Within this framework, the shrinkage parameter p quantifies the rate of erosion-induced geometric evolution and enables an accurate reconstruction of the experimental dexamethasone release profile. The solution reproduces both the initial fast-release phase and the extended depletion tail from which the characteristic times $t_{0.50}=264.3\mathrm{h}$ and $t_{0.90}=996.5\mathrm{h}$ are extracted, providing compact and physically meaningful indicators of the transition between early and late kinetic regimes. Overall, the approach offers a robust and interpretable description of drug release from shrinking polymeric systems and is directly applicable to the design of long-acting intravitreal therapies. Full article

This study examines the geometrically nonlinear free vibration behaviour of a clamped–clamped Euler–Bernoulli beam weakened by an open transverse crack. The crack is modelled through an equivalent rotational flexibility formulation grounded in fracture mechanics, while geometric nonlinearity is incorporated by accounting for mid-plane stretching associated with large vibration amplitudes. A reduced-order formulation is established using a Galerkin approach with eigenfunctions that explicitly depend on the presence of the crack, leading to a nonlinear eigenvalue problem in which the response is amplitude-dependent. The primary aim is to clarify how crack severity and vibration amplitude jointly influence the distribution of nonlinear bending stresses along the beam. Unlike much of the existing literature, which predominantly emphasises frequency–amplitude interactions, this work adopts a stress-oriented framework and offers a detailed characterisation of the spatial variation in the normalised stress field. The results indicate that bending stresses increase markedly as vibration amplitude grows, with the most pronounced effects occurring near the clamped ends where curvature is highest. Furthermore, deeper cracks significantly intensify stress concentrations due to the associated local reduction in stiffness. Overall, the findings provide enhanced physical understanding of the nonlinear dynamic behaviour of cracked beam structures and establish a useful foundation for evaluating structural integrity under conditions of large-amplitude vibration. Full article

Microwave imaging of objects embedded in multilayer cylindrical structures is of practical importance in applications where inaccessible targets are surrounded by a known inhomogeneous host. In such problems, incorporating the known multilayer structure into the background model can improve reconstruction accuracy and reduce the complexity of the inverse problem. This paper presents an efficient imaging method for dielectric objects embedded in two-dimensional multilayer cylindrical structures with arbitrarily shaped layer boundaries. The proposed approach integrates the contrast source inversion method with a recursive eigenfunction expansion technique for noncircular geometries. The known multilayer host is treated as the background medium, while the inversion is restricted to the embedded scatterers. The recursive formulation is derived to compute the inhomogeneous-background Green’s function and the required cell-integrated Green’s functions in a semi-analytical and discretization-free manner. Numerical results suggest that the method is capable of providing satisfactory reconstructions of embedded objects under various host configurations, including cases with a Perfect Electric Conductor (PEC) core. Comparisons with Method of Moments reference solutions confirm the accuracy of the forward modeling and the reliability of the inversion, while demonstrating a significant reduction in computational cost. Full article

The measurement of a probe impedance performed during eddy current inspections enables detection of flaws in electrically conductive materials. A correct interpretation of the measured impedance values constitutes a key aspect that determines the effectiveness of the inspections, and for this purpose, mathematical models are employed. Such models, which are becoming more and more frequently an integral part of eddy current measurement systems, enable carrying out the calculation of the probe impedance, through depicting the measurements being performed. What offer the shortest calculation time while maintaining high accuracy are analytical solutions. In this paper, to the best of the authors’ knowledge, this is the first time an analytical model of an eddy current probe placed over a small diameter cylinder containing a hole has been presented. The final formulas were obtained using the truncated region eigenfunction expansion (TREE) method, and then implemented in Matlab. The calculated values of the probe resistance and reactance were compared with the measurement results obtained for cylinders with a through defect. The tests were conducted on components made of several conductive materials with different geometric dimensions. The measurement error in all of the tests was small, i.e., it did not exceed 3% across the entire frequency range. The proposed solution can be used in defectoscopy for eddy current testing of tubes, pucks, washers, and any cylindrical elements. Full article