Search Results (60)

This paper investigates explicit coefficient-based estimates for a class of fractional Hermite functions defined through finite power series with Gamma-function coefficients. These functions may be viewed as a fractional Hermite-type family associated with the Caputo fractional derivative of order $\alpha \in (0,1]$. An explicit representation of the fractional derivative is obtained as a finite sum of monomials with computable Gamma coefficients. This representation is used to derive a preliminary uniform estimate on bounded intervals $[0,R]$ with an explicit constant depending on $\alpha$, n, and R. Consistency with the integer-order setting is established by showing that, when $\alpha =1$, the construction reduces to a Hermite-type polynomial family and the Caputo derivative coincides with the ordinary derivative. Explicit asymptotic formulas are obtained for the associated coefficient envelope as $R\to 0^{+}$ and $R\to \infty$. Numerical experiments up to degree $n=7$ show that the ratio between the coefficient envelope and the computed supremum norm remains below approximately $1.45$ for the tested parameter range. In addition, a weighted $L^2$ estimate is derived with respect to a fractional Gaussian-type weight, yielding an explicit coefficient-based bound. The estimates obtained in this work are preliminary in nature, being based on coefficient-wise majorization, and are not claimed to be optimal. Determining sharp constants and establishing genuine norm-comparison inequalities remain open problems. The results presented here provide a rigorous starting point for the study of explicit coefficient-based estimates for fractional Hermite functions and suggest several directions for future research in fractional approximation theory. Full article

In the present work, we introduce and study a new two-parameter generalization of Legendre-based Appell polynomials, defined through an explicit representation that unifies classical Legendre structures with the Appell polynomial framework. Starting from a generating function, we derive a three-term recurrence relation, a degree-lowering operator, an integro-partial degree-raising operator, and a corresponding integro-partial differential equation satisfied by the new family. A determinant representation is established via Cramer’s rule applied to the Cauchy-product expansion of the generating function. Several subfamilies of independent interest arise naturally as special cases, namely, Legendre-based Hermite–Frobenius–Euler polynomials, Legendre-based Miller–Lee polynomials, and both the probabilist’s and physicist’s variants of Legendre-based bi-variate Hermite polynomials. For each subfamily we record the corresponding recurrence relations, shift operators, differential equations, and determinant forms, and we illustrate the behavior of selected members through three-dimensional surface plots and real-root distribution diagrams. The framework presented here extends several constructions available in the recent literature and points to natural directions for future work, including connections with q-series, combinatorial identities, and symbolic-computation methods, which are outlined in the concluding section. Full article

Flash-flood and floodplain inundation simulations are highly sensitive to the spatiotemporal variability of convective rainfall, particularly during the initial runoff generation stage. However, coarse-resolution numerical weather prediction (NWP) forcing tends to smooth localized rainfall extremes, limiting its ability to accurately represent hydrological responses in low-relief floodplains. In this study, we couple a diffusion-enhanced radar nowcasting model, Diff_ConvLSTM, with a spatial resolution of 1 km and a temporal resolution of 6 min, to assess the hydrological value of high-resolution rainfall forcing over the middle Yangtze River floodplain. We introduce a monotone piecewise cubic Hermite interpolation scheme to ensure a stable transition from discrete high-frequency rainfall inputs to continuous hydrodynamic integration. Evaluation using a radar dataset from 2023 to 2024 shows that Diff_ConvLSTM better preserves intense convective echoes and rainband structures compared to the baseline ConvLSTM, increasing the Probability of Detection at the 40 dBZ threshold by 65.8%. A forcing-replacement experiment for the flood event on 30 June 2023 demonstrates that AI-based nowcasting rainfall forcing reduces peak-discharge underestimation, improves volumetric consistency, and produces inundation patterns that are closer to the observation-driven reference than those generated by low-resolution forecast forcing, although positive biases in inundation area and water depth persist. An additional event in 2024 confirms that the improvements are primarily reflected in discharge magnitude and flood volume representation, while enhancements in peak timing remain limited. Overall, the results illustrate both the added value and the remaining limitations of AI-enhanced nowcasting for hydrologically informed flood forecasting. Full article

In this work, the newly developed functional bootstrapping (FBT) for the Cheon–Kim–Kim–Song (CKKS) scheme is used for the first time to homomorphically evaluate an encrypted neural network. The advantage of FBT over previous approaches for the homomorphic evaluation of non-linear activation functions is that it combines bootstrapping and homomorphic function evaluation. For this purpose, FBT for CKKS is extended to be applied to real input values by evaluating the first order Hermite interpolation function not only on its interpolation points but on the entire domain $[0,1]$. For the sigmoid function, to respect the internal representation of negative values in CKKS and the convergence behaviour of trigonometric interpolation, a glueing of shifted and reflected sigmoid functions that is periodic and continuous is used as an input function for FBT. The experimental results yield an accuracy of 97.33% with a relative loss of $0\%$ compared to the Hermite plaintext counterpart that were obtained with a fully connected neural network with 100 hidden neurons on the MNIST test set at a security level of 128 bits. The current implementation required approximately 1.66 s per image (amortised time) and about 201 GB RAM. Full article

This article introduces and develops a comprehensive theory of the Mixed Degenerate Gould–Hopper–Appell Type Polynomials $MDGHA$-$TPs$, constructed by embedding an Appell factor into the framework of degenerate Gould–Hopper generating functions. Beginning with the generating function formulation, we derive explicit series representations, monomial-type operational identities, recurrence relations, and a determinantal form that encodes the algebraic structure of the family. Summation identities expressed via Stirling numbers of the first kind and addition-type formulas are established. A detailed numerical investigation of the zero distributions of these polynomials is then carried out, with graphical illustrations revealing symmetry patterns and geometric arrangements in the complex plane. Connections with classical sequences of Appell, Hermite, and Gould–Hopper are explored throughout. The article concludes with remarks on open problems including the orthogonality of the $MDGHA$-$TPs$ with respect to suitable weight functions, the asymptotic behaviour of their zeros as the degree tends to infinity, and potential applications to boundary-value problems in heat diffusion, perturbation expansions in quantum mechanics, and signal processing in non-homogeneous media. Full article

This paper presents a process-centric ontology for the semantic representation of cyber–physical systems (CPSs) within knowledge-based production planning (KPP). The approach integrates physical systems (PSs), cyber systems (CSs), and CPSs into a unified semantic model based on a three-layer classification. The ontology was implemented using OWL and integrated into a Neo4j-based graph architecture to support semantic querying and process modeling. The evaluation was conducted using prototypical manufacturing scenarios, including semiconductor and mechanical engineering domains. Validation included (i) consistency checking using the HermiT reasoner, (ii) execution of SPARQL queries for retrieving CPS-related process information, and (iii) integration into a three-stage planning model. The results show that the ontology enables consistent semantic representation and cross-domain querying of CPS-based production processes. The work provides a validated proof-of-concept and establishes a foundation for future research on ontology-based production systems. Full article

Deploying neural networks on encrypted data requires efficient evaluation of nonlinear activations, especially the ReLU function, without decryption. While the CKKS homomorphic encryption scheme supports packed arithmetic over approximate numbers efficiently, its approximate semantics make direct nonlinear evaluation difficult, and polynomial surrogates often introduce approximation error and non-discrete outputs. In this work, we present a task-specific, non-interactive construction for discrete ReLU evaluation in CKKS by combining modulus-switch-based discretization with interpolation-driven lookup-table (LUT) evaluation. We instantiate this design in two complementary schemes. The first uses trigonometric Hermite interpolation and functional bootstrapping to compute a discrete sign indicator, which is then combined with the encrypted input through conditional multiplication to obtain the ReLU output; this variant is compact and suitable for lightweight settings. The second uses iterative most-significant-bit (MSB) bootstrapping to support larger plaintext moduli and higher-precision regimes through repeated digit extraction. A common enabler of both schemes is a discretization step that maps approximate CKKS plaintexts to a finite integer representation; exactness in our setting therefore refers to exact evaluation over this discretized representation, while the deviation from the original CKKS plaintext is governed by the discretization error analyzed in Lemma 1. Experiments on encrypted MNIST inference and the accompanying LUT/storage analysis indicate that the proposed schemes preserve competitive accuracy relative to polynomial-approximation baselines while maintaining manageable auxiliary storage under the reported parameter settings. These results suggest that interpolation-based discrete activation is a promising alternative to polynomial approximation for selected CKKS-based encrypted inference tasks. Full article

In this paper, a new class of special polynomials, called the Sheffer-type general-$\lambda$-matrix polynomials, is introduced within the framework of the monomiality principle. This family is obtained by combining the structure of Sheffer sequences with the theory of general-$\lambda$ matrix polynomials, which leads to a unified formulation encompassing several polynomial families. Fundamental properties of the proposed polynomials are established, including their generating function, explicit series representation, summation formulas, quasi-monomial structure, differential relations, and determinant representation. The proposed framework addresses an important problem in the theory of special functions: the systematic construction of matrix-valued polynomial families that simultaneously generalize both classical scalar polynomials and existing matrix polynomial hierarchies. Such a unified structure is of broad significance, with applications in quantum mechanics (wave function expansions), mathematical physics (matrix differential equations and spectral problems), approximation theory, and the study of special functions in the matrix domain. Several hybrid forms of the proposed family are derived through appropriate choices of the defining functions, which yield polynomial subclasses related to classical families such as Hermite, Laguerre, Bessel, and Poisson–Charlier polynomials. These subclasses illustrate how the proposed framework provides a systematic approach for constructing and studying generalized polynomial structures. In each case, the matrix parameter L introduces a new layer of structural richness not present in the scalar setting, enabling the modelling of phenomena governed by matrix-valued spectral data. Furthermore, a numerical and graphical investigation of selected hybrid forms is carried out using Mathematica (version 14.3, 2025; Wolfram Research, Inc.). Surface plots, distributions of complex zeros, and real-zero patterns are presented for different parameter values, highlighting the influence of the parameters on the behavior and structural characteristics of the polynomials. Full article

In this paper, we give some new Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities for uniformly $\delta$-geometric convex functions. In addition, some limit bounds for Caputo–Fabrizio fractional integral operators are established as an application in the case of uniformly $\delta$-geometric convex functions. Some new examples and graphical representations are provided in order to illustrate the validity of our results. Full article

In this paper, we introduce, for the first time, a novel class of (center-radius order $(P,\mathrm{m})$-superquadratic interval-valued functions) $\mathfrak{cr}$-$(P,\mathrm{m})$-superquadratic $\mathcal{IVF}$s, and systematically investigate their fundamental structural properties. Building upon these properties, we establish new Jensen and Hermite–Hadamard ($\mathcal{HH}$) type inequalities, together with their fractional extensions formulated via Riemann–Liouville ($\mathcal{RL}$) fractional integral operators within the setting of interval calculus. The validity and sharpness of the derived results are illustrated through numerical examples and graphical representations. Moreover, the theoretical developments are further enriched by applications in information theory, leading to meaningful generalizations and notable improvements over several existing results reported in the literature. Full article

Healthcare fragmentation—characterized by poor coordination among independently operating organizations—systematically degrades care quality while escalating costs. While healthcare delivery inherently operates as a System of Systems (SoS), existing approaches lack semantic rigor to bridge governance principles with implementable architectures, and digital engineering paradigms remain disconnected from formal representations of regulatory constraints and organizational interdependencies. This paper presents a comprehensive Web Ontology Language (OWL 2 DL)-based ontology integrating structural, behavioral, and regulatory dimensions of healthcare SoS into a unified, computationally tractable framework. Developed following the Methontology engineering methodology and validated using the HermiT reasoner, the ontology formalizes constituent system categories through functional decomposition, establishes an interaction taxonomy distinguishing intra-category coordination from inter-category integration, and introduces the Covenant class as a novel governance mechanism. The covenant embeds legal frameworks (HIPAA, GDPR), interoperability protocols (FHIR, HL7), and technical standards (SNOMED, LOINC, ICD-11, ISO) as first-class ontological entities with explicit relationships to interaction properties. Governance enforcement is operationalized through a layered validation architecture comprising SWRL rules for deductive compliance checking, SHACL shapes for structural constraint validation, and OWL equivalentClass axioms for automated conflict detection. The ontology is further validated through four operational scenarios that demonstrate automated consent validation, standards compliance verification, protocol interoperability checking, and temporal compliance with conflict detection, alongside extended SPARQL queries that reveal constituent system landscapes, standards coverage, interaction networks, and topological properties through node degree calculation, hub identification, and network density analysis. The ontology enables pre-implementation governance assessments, evidence-based policy simulation, digital twin implementations with continuous compliance monitoring, and resilience planning through network analysis, transforming governance from reactive compliance checking to proactive coordination engineering. Full article

The conventional time-dependent perturbation theory in quantum mechanics is established within the framework of Hermitian Hamiltonians, applicable for describing quantum transitions and associated energy level responses in such systems. However, this theory has fundamental limitations when applied to non-Hermitian systems. Consequently, researchers have systematically extended time-dependent perturbation theory to non-Hermitian systems, establishing a corresponding mature framework. Building on this foundation, this study extends the theory to investigate the transition dynamics induced by non-Hermitian interactions in non-Hermitian Hamiltonian systems. We employ a biorthogonal basis representation for a three-level non-Hermitian system. This work investigates a system comprising an unperturbed static non-Hermitian Hamiltonian and a periodically driven time-dependent perturbation Hamiltonian. Taking the three-level system as a concrete example, we combine analytical methods with numerical simulations to solve and analyze its dynamical evolution equations. These complementary approaches reveal that when system parameters complete a full cycle around an exceptional point, the transitional behavior exhibits specific evolutionary patterns. In this system, quantum transition probabilities exhibit significant asymmetry and non-conservation that depend on the initial and final states, revealing inherent directional characteristics in the dynamical process. Furthermore, for a three-level, periodically driven non-Hermitian system with time-dependent perturbations, this asymmetry is even more pronounced, manifesting as a distinct disparity between forward and reverse transition probabilities. The periodic driving actively amplifies the asymmetry in the transition process. By designing the perturbation spectrum, selective manipulation of specific quantum states can be achieved. Moreover, transition probabilities can be significantly enhanced under resonance conditions, while non-Hermiticity further breaks the system’s inherent symmetry, leading to substantial amplification of transitions in a single direction. By precisely tuning the drive frequency, interactions between specific coupling channels can be selectively enhanced or suppressed. The amplification of channel asymmetry by non-Hermitian properties provides a novel mechanism for directional control of quantum states and opens new pathways for realizing related quantum technologies. Full article

Improving the simulation efficiency of the spectral representation method (SRM) for nonstationary fluctuating wind fields has attracted considerable attention. To this end, this study proposes a method based on proper orthogonal decomposition (POD) decoupling and Spline interpolation to enhance computational efficiency. This method selects a limited number of interpolation points in the time-frequency domain of the evolutionary power spectral density (EPSD) for Cholesky decomposition, utilizes the proper orthogonal decomposition (POD) technique to achieve time-frequency decoupling of the spectral matrix, and employs Spline interpolation but not the traditional Hermite-interpolation to reconstruct the complete time-frequency functions, thereby enabling the rapid synthesis of wind-velocity time histories via the FFT. Then, the wind field on a three-span frame lightning-rod structure is taken as an example to validate the reliability of the proposed method. The influences of the modal order and the number of time-frequency interpolation points on both simulation efficiency and error are investigated, and comparisons are given with the Hermite-interpolation-based method. The results indicate that the simulation efficiency is governed primarily by the modal order, and the method with Spline interpolation shows higher computational efficiency and accuracy because it can satisfy accuracy requirements at a lower modal order. Finally, a rational truncation criterion based on the cumulative energy ratio of at least 99.9% is suggested to determine the optimal modal order, thereby achieving a balance between accuracy and computational efficiency. Full article

The purpose of this research is to develop a set of Hermite–Hadamard–Mercer-type inequalities that involve different types of fractional integral operators such as classical Riemann–Liouville fractional integral operators. Furthermore, some fractional integral inequalities are obtained for three-times differentiable convex functions with respect to the right-hand side of the Hermite–Hadamard–Mercer-type inequality. Moreover, several new results regarding Young’s inequality, bounded function and L-Lipschitzian function are deduced. The paper presents additional remarks and comments on the results to make sense of them. To illustrate the key findings, graphical representations are provided, and applications involving special means, midpoint formula, q-digamma function and modified Bessel function are presented to demonstrate the practical utility of the derived inequalities. Full article

This article investigates fractional Hermite–Hadamard integral inequalities through the framework of Caputo fractional derivatives and MET-$(p,s)$-convex functions. In particular, we introduce new modifications to two classical fractional extensions of Hermite–Hadamard-type inequalities, formulated for both MET-$(p,s)$-convex functions and logarithmic $(p,s)$-convex functions. Moreover, we obtain enhancements of inequalities like the Hermite–Hadamard, midpoint, and Fejér types for two extended convex functions by employing the Caputo fractional derivative. The research presents a numerical example with graphical representations to confirm the correctness of the obtained results. Full article