Search Results (8)

This work presents a comprehensive mathematical framework for symmetrized neural network operators operating under the paradigm of fractional calculus. By introducing a perturbed hyperbolic tangent activation, we construct a family of localized, symmetric, and positive kernel-like densities, which form the analytical backbone for three classes of multivariate operators: quasi-interpolation, Kantorovich-type, and quadrature-type. A central theoretical contribution is the derivation of the Voronovskaya–Santos–Sales Theorem, which extends classical asymptotic expansions to the fractional domain, providing rigorous error bounds and normalized remainder terms governed by Caputo derivatives. The operators exhibit key properties such as partition of unity, exponential decay, and scaling invariance, which are essential for stable and accurate approximations in high-dimensional settings and systems governed by nonlocal dynamics. The theoretical framework is thoroughly validated through applications in signal processing and fractional fluid dynamics, including the formulation of nonlocal viscous models and fractional Navier–Stokes equations with memory effects. Numerical experiments demonstrate a relative error reduction of up to 92.5% when compared to classical quasi-interpolation operators, with observed convergence rates reaching $\mathcal{O}\left(n^{-1.5}\right)$ under Caputo derivatives, using parameters $\lambda =3.5$, $q=1.8$, and $n=100$. This synergy between neural operator theory, asymptotic analysis, and fractional calculus not only advances the theoretical landscape of function approximation but also provides practical computational tools for addressing complex physical systems characterized by long-range interactions and anomalous diffusion. Full article

This article introduces the 2-variable q-truncated exponential–Appell (q-trunc. exp. Appell) polynomials and investigates their fundamental properties. Specific results are derived for the q-trunc. exp. Appell family along with their graphical representations which contribute to advancing the understanding of q-series and q-special functions. Potential applications of these polynomials span various disciplines, including combinatorics (such as partition theory and combinatorial identities), number theory (such as q-analogues of classical number-theoretic functions), and mathematical physics (such as in quantum groups and statistical mechanics). This study concludes with the introduction of the 2-variable q-trunc. exp. $\lambda$-polynomials, thereby broadening the scope and relevance of this research. Full article

Inferences on the location parameter $\lambda$ in location-scale families can be carried out using Studentized statistics, i.e., considering estimators $\tilde{\lambda }$ of $\lambda$ and $\tilde{\delta }$ of the nuisance scale parameter $\delta$, in a statistic $T=g(\tilde{\lambda },\tilde{\delta })$ with a sampling distribution that does not depend on $(\lambda ,\delta )$. If both estimators are independent, then T is an externally Studentized statistic; otherwise, it is an internally Studentized statistic. For the Gaussian and for the exponential location-scale families, there are externally Studentized statistics with sampling distributions that are easy to obtain: in the Gaussian case, Student’s classic t statistic, since the sample mean $\tilde{\lambda }=\overline{X}$ and the sample standard deviation $\tilde{\delta }=S$ are independent; in the exponential case, the sample minimum $\tilde{\lambda }=X_{1:n}$ and the sample range $\tilde{\delta }=X_{n:n}-X_{1:n}$, where the latter is a dispersion estimator, which are independent due to the independence of spacings. However, obtaining the exact distribution of Student’s statistic in non-Gaussian populations is hard, but the consequences of assuming symmetry for the parent distribution to obtain approximations allow us to determine if Student’s statistic is conservative or liberal. Moreover, examples of external and internal Studentizations in the asymmetric exponential population are given, and an ANalysis Of Spacings (ANOSp) similar to an ANOVA in Gaussian populations is also presented. Full article

Yongle Atoll was the largest atoll in the Xisha Islands of the South China Sea, and it was a coral reef ecosystem with important ecological and economic values. In order to better protect and manage the coral reef fish resources in Yongle Atoll, we analyzed field survey data from artisanal fishery, catches, and underwater video from 2020 to 2022 and combined historical research to explore the changes in fish species composition and community structure in Yongle Atoll over the past 50 years. The results showed that a total of 336 species of fish were found on Yongle Atoll, belonging to 17 orders and 60 families. Among them, Perciformes had the most fish species with 259 species accounting for 77.08% of the total number of species. The number of fish species in the coral reef of Yongle Atoll was exponentially correlated with its corresponding maximum length and significantly decreases with its increase. The fish community structure of Yongle Atoll changed, and the proportion of large carnivorous fish decreased significantly, while the proportion of small-sized and medium-sized fish increased. At the same time, Yongle Atoll has 18 species of fish listed on the IUCN Red List, 15 of which are large fish. The average taxonomic distinctness (Delta+, Δ+) and the variation taxonomic distinctness (Lambda+, Λ+) in 2020–2022 were lower than the historical data, and the number of fish orders, families, and genera in Yongle Atoll has decreased significantly, which indicates that the current coral reef fish species in Yongle Atoll have closer relatives and higher fish species uniformity. In addition, the similarity of fish species in Yongle Atoll was relatively low at various time periods, further proving that the fish community structure has undergone significant variation. In general, due to multiple impacts, such as overfishing, fishing methods, environmental changes, and habitat degradation, the fish species composition of Yongle Atoll may have basically evolved from carnivorous to herbivorous, from large fish to small fish, and from complexity to simplicity, leaving Yongle Atoll in an unstable state. Therefore, we need to strengthen the continuous monitoring of the coral reef ecosystem in Yongle Atoll to achieve the protection and restoration of its ecological environment and fishery resources, as well as sustainable utilization and management. Full article

In this work, the case of a Cox Process with Folded Normal Intensity (CP-FNI), in which the intensity is given by $\mathrm{\Lambda }\left(t\right)=\left|Z\right(t\left)\right|$, where $Z\left(t\right)$ is a stationary Gaussian process, is studied. Here, two particular cases are dealt with: (i) when the process $Z\left(t\right)$ constitutes a family of independent random variables and with a common probability law $\mathrm{N}(0,1)$, and (ii) the case in which $Z\left(t\right)$ is a second order stationary process, with exponential type covariance function. In these cases, we observe that the properties of the Gaussian process $Z\left(t\right)$ are naturally transferred to the intensity $\mathrm{\Lambda }\left(t\right)$ and that very analytical results are achievable from the analytical point of view for the point process $N\left(t\right)$. Finally, some simulations are presented in order to appreciate what type of counting phenomena can be modeled by these cases of CP-FNI. In particular, it is interesting to see how the trajectories show a tendency of the events to be grouped in certain periods of time, also leaving long periods of time without the occurrence of events. Full article

This paper systematically presents the $\lambda$-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the $\lambda$-deformed case: $\lambda$-convexity, $\lambda$-conjugation, $\lambda$-biorthogonality, $\lambda$-logarithmic divergence, $\lambda$-exponential and $\lambda$-mixture families, etc. In particular, $\lambda$-deformation unifies Tsallis and Rényi deformations by relating them to two manifestations of an identical $\lambda$-exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the $\lambda$-exponential family, in turn, coincides with the $\lambda$-mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus, $\lambda$-deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry. Full article

Hydraulic conductivity k as a function of void ratio e and particle diameter for silty sediment was experimentally investigated, and an empirical formula for the estimation of hydraulic conductivity was proposed. Seepage resistance for flow in silty sediment was deliberated. Based on the findings of the study, it was concluded that hydraulic conductivity k could be expressed as an exponential function of void ratio e and median particle diameter d₅₀ {3.1 μm < d₅₀ < 87 μm and 0.26 < e < 4}. It was further found that the formula of seepage resistance factor (λ), a form of friction factor, varies linearly with Reynolds number ($R_e)$ for silty sediments. A family of such $\lambda$-$R_e$ curves for various particle diameter d₅₀ is presented. Full article

We consider translates of functions in $L^2({\mathbb{R}}^d)$ along an irregular set of points, that is, ${\{\varphi (·-{\lambda }_k)\}}_{k\in \mathbb{Z}}$ —where $\varphi$ is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform. Full article