Search Results (33)

This article investigates the lower bounds of analytic functions defined by absolutely convergent Dirichlet series in the left half-plane and establishes conditions under which such functions exhibit the pits property. The study extends classical results for entire Dirichlet series and lacunary power series by refining assumptions on the sequence of exponents and resolving gaps in earlier proofs found in the literature. Central to the analysis is the introduction of a modified function $k\left(\sigma \right)$, whose behavior determines the size of exceptional sets surrounding the zeros of the series. Under suitable growth conditions on the exponents, the authors prove that outside small disks centered at the zeros, the modulus of the Dirichlet series admits explicit lower bounds involving its maximum term. Several auxiliary lemmas provide sharp estimates for the maximum modulus, the distribution of zeros, and the behavior of truncated Dirichlet polynomials. The main theorem demonstrates that the pits property holds uniformly in vertical strips approaching the imaginary axis. The paper concludes with a discussion of how the imposed step-size condition on exponents might be weakened, formulating a conjecture regarding a more general condensation index. Full article

This paper studies the algebraic properties of little Hankel operators on Hardy–Sobolev spaces $H_s^2$, focusing on a notion of commutativity defined via adjoint products. For symbols $\phi$ and $\psi$, whose co-analytic parts are trigonometric polynomials, we consider the condition ${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}\mathrm{on}{H}_s^2.$ It is shown that this adjoint-product commutativity holds if and only if the co-analytic parts of the symbols are real scalar multiples of one another. As a consequence, the commutant of a nonzero Hankel operator on $H_s^2$, within the class of Hankel operators whose co-analytic symbols are trigonometric polynomials, is one-dimensional over $\mathbb{R}$. The proof relies on a direct coefficient analysis exploiting the finite Hankel structure induced by polynomial symbols. The result applies uniformly to all Sobolev exponents $s\ge 0$, including the classical Hardy space $s=0$ and the Dirichlet case $s=1/2$. Full article

This study introduces a high-order numerical scheme for solving 1D second-order hyperbolic telegraph equations (HTEs) with variable coefficients. We employ a generalized temporal discretization (TD) of order p via the Rothe approach, combined with a spatial spectral collocation (SCM) method using generalized shifted Jacobi polynomials (GSJPs). By utilizing a Galerkin-type basis that structurally satisfies homogeneous boundary conditions (HBCs)—including Dirichlet or Neumann types—we achieve a global error bound of $\mathcal{O}({(\Delta \tau )}^p+N^{-s})$, where $\Delta \tau$ denotes the temporal step size and s represents the spatial regularity of the exact solution (ExaS). The proposed algorithm, Rothe-GSJP, allows for an optimal balance between the temporal and spatial parameters, minimizing computational effort for high-precision engineering applications such as Phase-Locked Loop (PLL) modeling. Numerical experiments performed on an i9-10850 workstation show that the scheme always reaches the machine precision floor of ${10}^{-16}$. While the framework supports temporal orders up to $p=6$, the results indicate that $p\in \{2,3,4\}$ provides an optimal balance between high-order precision and absolute stability. The Rothe-GSJP method proves to be a robust, efficient, and highly accurate alternative to traditional solvers for hyperbolic systems. Full article

We investigate a random field of mutually dependent random variables (“spins”), indexed by a finite one-dimensional lattice, called in physical sciences the one-dimensional Ising model, in which the random variables can take only ±1 values (see the text for a precise definition). One of the couplings, termed a “bond,” that describes the mutual influence of two adjacent random variables is altered—it does not equal the others, thereby introducing a single “defect” bond. This defect bond represents a localised perturbation within an otherwise uniform system. Utilising the recurrence relations of Chebyshev polynomials and the bijective map between the number of spins and the polynomial index, we present a new method for calculations and systematically explore, using it, the system’s properties across different chain lengths and boundary conditions. As an application, we derive analytical expressions for the dependence of the average values of the random variables on their position within the chain, which we refer to as the “local magnetisation profile”. From the findings related to the system with a defect bond, we present a novel result for this profile under free (Dirichlet) boundary conditions and re-derive the corresponding result for antiperiodic boundary conditions. Full article

In freeform optical metrology, wavefront fitting over non-circular apertures is hindered by the loss of Zernike polynomial orthogonality and severe sampling grid distortion inherent in standard conformal mappings. To address the resulting numerical instability and fitting bias, we propose a unified framework curve-shortening flow (CSF)-guided progressive quasi-conformal mapping (CSF-QCM), which integrates geometric boundary evolution with topology-aware parameterization. CSF-QCM first smooths complex boundaries via curve-shortening flow, then solves a sparse Laplacian system for harmonic interior coordinates, thereby establishing a stable diffeomorphism between physical and canonical domains. For doubly connected apertures, it preserves topology by computing the conformal modulus via Dirichlet energy minimization and simultaneously mapping both boundaries. Benchmarked against state-of-the-art methods (e.g., Fornberg, Schwarz–Christoffel, and Ricci flow) on representative irregular apertures, CSF-QCM suppresses area distortion and restores discrete orthogonality of the Zernike basis, reducing the Gram matrix condition number from >900 to <8. This enables high-precision reconstruction with RMS residuals as low as $3\times {10}^{-3}\lambda$ and up to 92% lower fitting errors than baselines. The framework provides a unified, computationally efficient, and numerically stable solution for wavefront reconstruction in complex off-axis and freeform optical systems. Full article

In this paper, we develop a concise differential–potential framework for the functions of a generalized quaternionic variable in the two-parameter algebra ${\mathbb{H}}_{\alpha ,\beta }$, with $\alpha ,\beta \in \mathbb{R}\setminus \left\{0\right\}$. Starting from left/right difference quotients, we derive complete Cauchy–Riemann (CR) systems and prove that, away from the null cone where the reduced norm N vanishes, these first-order systems are necessary and, under ${\mathcal{C}}^1$ regularity, sufficient for left/right differentiability, thereby linking classical one-dimensional calculus to a genuinely four-dimensional setting. On the potential theoretic side, the Dirac factorization ${\Delta }_{\alpha ,\beta }=\overline{\mathcal{D}}\mathcal{D}=\mathcal{D}\overline{\mathcal{D}}$ shows that each real component of a differentiable mapping is ${\Delta }_{\alpha ,\beta }$-harmonic, yielding a clean second-order theory that separates the elliptic (Hamiltonian) and split (coquaternionic) regimes via the principal symbol. In the classical case $(\alpha ,\beta )=(-1,-1)$, we present a Poisson-type representation solving a model Dirichlet problem on the unit ball $B\subset {\mathbb{R}}^4$, recovering mean-value and maximum principles. For computation and symbolic verification, real $4\times 4$ matrix models for left/right multiplication linearize the CR systems. Examples (polynomials, affine CR families, and split-signature contrasts) illustrate the theory, and the outlook highlights boundary integral formulations, Green kernel constructions, and discretization strategies for quaternionic PDEs. Full article

The aim of this work was to construct explicit expressions for the summation of Dirichlet Beta functions with odd arguments and Zeta functions with even arguments. In the established literature, this is typically done using Fourier series expansions or Bernoulli numbers and polynomials. Here, instead, we achieve our goal by employing tools from probability: specifically, we introduce a generalisation of a technique based on multiple integrals and the algebra of random variables. This also allows us to increase the number of nested integrals and Cauchy random variables involved. Another key contribution is that, by generalising the exponent of Cauchy random variables, we obtain an original proof of the reflection formulae for the Digamma and Trigamma functions. These probabilistic proofs crucially utilise the Mellin transform to compute the integrals needed to determine probability density functions. It is noteworthy that, while understanding the presented topic requires knowledge of the rules for calculating multiple integrals (Fubini’s Theorem) and the algebra of continuous random variables, these are concepts commonly acquired by second-year university students in STEM disciplines. Our study thus offers new perspectives on how the mathematical functions considered relate and shows the significant role of probabilistic methods in promoting comprehension of this research area, in a way accessible to a broad and non-specialist audience. Full article

This paper investigates the derivation of global shape functions in T-meshed quadrilateral patches through transfinite interpolation and local elimination. The same shape functions may be alternatively derived starting from a background tensor product of Lagrange polynomials and then imposing linear constraints. Based on the nodal points of the T-mesh, which are associated with the primary degrees of freedom (DOFs), all the other points of the background grid (i.e., the secondary DOFs) are interpolated along horizontal and vertical stations (isolines) of the tensor product, and thus, linear relationships are derived. By implementing these constraints into the original formula/expression, global shape functions, which are only associated with primary DOFs, are created. The quality of the elements is verified by the numerical solution of a typical potential problem of second order, with boundary conditions of Dirichlet and Neumann type. Full article

The paper advances a new technique for constructing the exceptional differential polynomial systems (X-DPSs) and their infinite and finite orthogonal subsets. First, using Wronskians of Jacobi polynomials (JPWs) with a common pair of the indexes, we generate the Darboux–Crum nets of the rational canonical Sturm–Liouville equations (RCSLEs). It is shown that each RCSLE in question has four infinite sequences of quasi-rational solutions (q-RSs) such that their polynomial components from each sequence form a X-Jacobi DPS composed of simple pseudo-Wronskian polynomials (p-WPs). For each p-th order rational Darboux Crum transform of the Jacobi-reference (JRef) CSLE, used as the starting point, we formulate two rational Sturm–Liouville problems (RSLPs) by imposing the Dirichlet boundary conditions on the solutions of the so-called ‘prime’ SLE (p-SLE) at the ends of the intervals (−1, +1) or (+1, ∞). Finally, we demonstrate that the polynomial components of the q-RSs representing the eigenfunctions of these two problems have the form of simple p-WPs composed of p Romanovski–Jacobi (R-Jacobi) polynomials with the same pair of indexes and a single classical Jacobi polynomial, or, accordingly, p classical Jacobi polynomials with the same pair of positive indexes and a single R-Jacobi polynomial. The common, fundamentally important feature of all the simple p-WPs involved is that they do not vanish at the finite singular endpoints—the main reason why they were selected for the current analysis in the first place. The discussion is accompanied by a sketch of the one-dimensional quantum-mechanical problems exactly solvable by the aforementioned infinite and finite EOP sequences. Full article

This paper extends the well-known transfinite interpolation formula, which was developed in the late 1960s by the applied mathematician William Gordon at the premises of General Motors as an extension of the pre-existing Coons interpolation formula. Here, a conjecture is formulated, which claims that the meaning of the involved blending functions can be enhanced, such that it includes any linear independent and complete set of functions, including piecewise-linear, trigonometric functions, Bernstein polynomials, B-splines, and NURBS, among others. In this sense, NURBS-based isogeometric analysis and aspects of T-splines may be considered as special cases. Applications are provided to illustrate the accuracy in the interpolation through the L₂ error norm of closed-formed functions prescribed at the nodal points of the transfinite patch, which represent the solution of partial differential equations under boundary conditions of the Dirichlet type. Full article

This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equations (RCSLEs) with quasi-rational solutions (q-RSs) were obtained by applying rational Rudjak–Zakhariev transformations (RRZTs) to the Jacobi equation re-written in the canonical form. The presented analysis was focused on the RRZTs leading to the canonical form of the Heun equation. It was demonstrated that the latter equation preserves its form under the second-order Darboux–Crum transformation. The associated Sturm–Liouville problems (SLPs) were formulated for the so-called ‘prime’ SLEs solved under the Dirichlet boundary conditions (DBCs). It was proven that one of the two X₁-Jacobi DPSs composed of Heun polynomials contains both the X₁-Jacobi orthogonal polynomial system (OPS) and the finite EOP sequence composed of the pseudo-Wronskian transforms of Romanovski–Jacobi (R-Jacobi) polynomials, while the second analytically solvable Heun equation does not have the discrete energy spectrum. The quantum-mechanical realizations of the developed formalism were obtained by applying the Liouville transformation to each of the SLPs formulated in such a way. Full article

Conventional approaches for the structural health monitoring of infrastructures often rely on physical sensors or targets attached to structural members, which require considerable preparation, maintenance, and operational effort, including continuous on-site adjustments. This paper presents an image-driven hybrid structural analysis technique that combines digital image processing (DIP) and regression analysis with a continuum point cloud method (CPCM) built on a particle-based strong formulation. Polynomial regressions capture the boundary shape change due to the structural loading and precisely identify the edge and corner coordinates of the deformed structure. The captured edge profiles are transformed into essential boundary conditions. This allows the construction of a strongly formulated boundary value problem (BVP), classified as the Dirichlet problem. Capturing boundary conditions from the digital image is novel, although a similar approach was applied to the point cloud data. It was shown that the CPCM is more efficient in this hybrid simulation framework than the weak-form-based numerical schemes. Unlike the finite element method (FEM), it can avoid aligning boundary nodes with regression points. A three-point bending test of a rubber beam was simulated to validate the developed technique. The simulation results were benchmarked against numerical results by ANSYS and various relevant numerical schemes. The technique can effectively solve the Dirichlet-type BVP, yielding accurate deformation, stress, and strain values across the entire problem domain when employing a linear strain model and increasing the number of CPCM nodes. In addition, comparative analysis with conventional displacement tracking techniques verifies the developed technique’s robustness. The proposed technique effectively circumvents the inherent limitations of traditional monitoring methods resulting from the reliance on physical gauges or target markers so that a robust and non-contact solution for remote structural health monitoring in real-scale infrastructures can be provided, even in unfavorable experimental environments. Full article

The Hurwitz zeta-function $\zeta (s,\alpha )$, $s=\sigma +it$, with parameter $0<\alpha ⩽1$ is a generalization of the Riemann zeta-function $\zeta \left(s\right)$ ($\zeta (s,1)=\zeta \left(s\right)$) and was introduced at the end of the 19th century. The function $\zeta (s,\alpha )$ plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function $\zeta (s,\alpha )$ is the main example of zeta-functions without Euler’s product (except for the cases $\alpha =1$, $\alpha =1/2$), and its value distribution is governed by arithmetical properties of $\alpha$. For the majority of zeta-functions, $\zeta (s,\alpha )$ for some $\alpha$ is universal, i.e., its shifts $\zeta (s+i\tau ,\alpha )$, $\tau \in \mathbb{R}$, approximate every analytic function defined in the strip $\{s:1/2<\sigma <1\}$. For needs of effectivization of the universality property for $\zeta (s,\alpha )$, the interval for $\tau$ must be as short as possible, and this can be achieved by using the mean square estimate for $\zeta (\sigma +it,\alpha )$ in short intervals. In this paper, we obtain the bound $O\left(H\right)$ for that mean square over the interval $[T-H,T+H]$, with $T^{27/82}⩽H⩽T^{\sigma }$ and $1/2<\sigma ⩽7/12$. This is the first result on the mean square for $\zeta (s,\alpha )$ in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for $\zeta (s,\alpha )$ and other zeta-functions in short intervals. Full article

The current study discusses a novel approach for numerically solving MTVO-TFDWEs under various conditions, such as IBCs and DBCs. It uses a class of GSJPs that satisfy the given conditions (IBCs or DBCs). One of the important parts of our method is establishing OMs for Ods and VOFDs of GSJPs. The second part is using the SCM by utilizing these OMs. This algorithm enables the extraction of precision and efficacy in numerical solutions. We provide theoretical assurances of the treatment’s efficacy by validating its convergent and error investigations. Four examples are offered to clarify the approach’s practicability and precision; in each one, the IBCs and DBCs are considered. The findings are compared to those of preceding studies, verifying that our treatment is more effective and precise than that of its competitors. Full article

This study proposes a novel hybrid simulation technique for analyzing structural deformation and stress using light detection and ranging (LiDAR)-scanned point cloud data (PCD) and polynomial regression processing. The method estimates the edge and corner points of the deformed structure from the PCD. It transforms into a Dirichlet boundary condition for the numerical simulation using the particle difference method (PDM), which utilizes nodes only based on the strong formulation, and it is advantageous for handling essential boundaries and nodal rearrangement, including node generation and deletion between analysis steps. Unlike previous studies, which relied on digital images with attached targets, this research uses PCD acquired through LiDAR scanning during the loading process without any target. Essential boundary condition implementation naturally builds a boundary value problem for the PDM simulation. The developed hybrid simulation technique was validated through an elastic beam problem and a three-point bending test on a rubber beam. The results were compared with those of ANSYS analysis, showing that the technique accurately approximates the deformed edge shape leading to accurate stress calculations. The accuracy improved when using a linear strain model and increasing the number of PDM model nodes. Additionally, the error that occurred during PCD processing and edge point extraction was affected by the order of polynomial regression equation. The simulation technique offers advantages in cases where linking numerical analysis with digital images is challenging and when direct mechanical gauge measurement is difficult. In addition, it has potential applications in structural health monitoring and smart construction involving machine leading techniques. Full article