Search Results (245)

Humanoid robots have attracted considerable attention for their anthropomorphic structure, extended workspace, and versatile capabilities. This paper presents a novel humanoid upper-body robotic system comprising a pair of 8-degree-of-freedom (DOF) arms, a 3-DOF head, and a 3-DOF torso—yielding a 22-DOF architecture inspired by human biomechanics and implemented via standardized hollow joint modules. To overcome the critical reliance of zeroing neural network (ZNN)-based trajectory tracking on the Jacobian matrix derivative, we propose an integration-enhanced matrix derivative observer (IEMDO) that incorporates nonlinear feedback and integral correction. The observer is theoretically proven to ensure asymptotic convergence and enables accurate, real-time estimation of matrix derivatives, addressing a fundamental limitation in conventional ZNN solvers. Workspace analysis reveals that the proposed design achieves an 87.7% larger total workspace and a remarkable 3.683-fold expansion in common workspace compared to conventional dual-arm baselines. Furthermore, the observer demonstrates high estimation accuracy for high-dimensional matrices and strong robustness to noise. When integrated into the ZNN controller, the IEMDO achieves high-precision trajectory tracking in both simulation and real-world experiments. The proposed framework provides a practical and theoretically grounded approach for redundant humanoid arm control. Full article

In this paper, voxel-based Asymptotic Homogenization (AH) is employed to calculate the thermal expansion and thermal conductivity characteristics of lattice materials that have a Representative Volume Element (RVE) with non-orthogonal periodic bases. The non-orthogonal RVE of the cellular lattice is discretized using voxel elements (iso-parametric hexahedral element, on a cartesian grid). A homogenization framework is developed in python that uses a fast-nearest neighbor algorithm to approximate the (non-orthogonal) periodic boundary conditions of the discretized RVE. Validation studies are performed where results of the homogenized Thermal Expansion Coefficient (TEC) and thermal conduction performed in this paper are compared with results generated by commercially available software. These included comparison with the results for (a) bi-material unidirectional composite with orthogonal RVE cell envelope; (b) bi-material hexagon lattice with orthogonal cell envelope; (c) bi-material hexagon lattice with non-orthogonal cell envelope; and (d) bi-material square lattice. A novel approach of visualizing the contribution of each voxel towards the individual terms within the homogenized thermal conductivity matrix is presented, which is necessary to mitigate any potential errors arising from the numerical model. Additionally, the effect of the thermal expansion and thermal conductivity for bi-material hexagon lattice (orthogonal and non-orthogonal RVE cell envelope) are presented for varying internal cell angles and all permutations of material assignments for a relative density of 0.3. It is found that when comparing the non-orthogonal RVE with the Orthogonal RVE as a reference model, the numerical error due to approximating the periodic boundary condition for the non-orthogonal bi-material hexagon is generally less than 2% as the numerical error is pseudo-cyclically dependent on the discretization along the cartesian axis. Full article

We introduce a modern methodology for constructing global analytical approximations of special functions over their entire domains. By integrating the traditional method of matching asymptotic expansions—enhanced with Padé approximants—with differential evolution optimization, a modern machine learning technique, we achieve high-accuracy approximations using elegantly simple expressions. This method transforms non-elementary functions, which lack closed-form expressions and are often defined by integrals or infinite series, into simple analytical forms. This transformation enables deeper qualitative analysis and offers an efficient alternative to existing computational techniques. We demonstrate the effectiveness of our method by deriving an analytical expression for the Fermi gas pressure that has not been previously reported. Additionally, we apply our approach to the one-loop correction in thermal field theory, the synchrotron functions, common Fermi–Dirac integrals, and the error function, showcasing superior range and accuracy over prior studies. Full article

In composites, fiber–matrix thermal mismatch induces stress heterogeneity that is beyond the resolution of macroscopic approaches. The asymptotic expansion homogenization method is used to create a multi-scale thermo-mechanical coupling model that predicts the elastic modulus, thermal expansion coefficients, and thermal conductivity of ceramic matrix composites at both the macro- and micro-scales. These predictions are verified to be accurate with a maximum relative error of 9.7% between the measured and predicted values. The multi-scale analysis method is then used to guide the vane’s thermal stress analysis, and a macro–meso–micro multi-scale model is created. The thermal stress distribution and stress magnitudes of the guide vane under a transient high-temperature load are investigated. The results indicate that the temperature and thermal stress distributions of the guide vane under the homogenization and lamination theory models are rather comparable, and the locations of the maximum thermal stress are predicted to be reasonably close to one another. The homogenization model allows for the rapid and accurate prediction of the guide vane’s thermal stress distribution. When compared to the macro-scale stress values, the meso-scale predicted stress levels exhibit excellent accuracy, with an inaccuracy of 11.7%. Micro-scale studies reveal significant stress concentrations at the fiber–matrix interface, which is essential for the macro-scale fatigue and fracture behavior of the guide vane. Full article

We address the problem of the distributed computation of arbitrary functions of two correlated sources, $X_1$ and $X_2$, residing in two distributed source nodes, respectively. We exploit the structure of a computation task by coding source characteristic graphs (and multiple instances using the n-fold OR product of this graph with itself). For regular graphs and general graphs, we establish bounds on the optimal rate—characterized by the chromatic entropy for the n-fold graph products—that allows a receiver for asymptotically lossless computation of arbitrary functions over finite fields. For the special class of cycle graphs (i.e., 2-regular graphs), we establish an exact characterization of chromatic numbers and derive bounds on the required rates. Next, focusing on the more general class of d-regular graphs, we establish connections between d-regular graphs and expansion rates for n-fold graph products using graph spectra. Finally, for general graphs, we leverage the Gershgorin Circle Theorem (GCT) to provide a characterization of the spectra, which allows us to derive new bounds on the optimal rate. Our codes leverage the spectra of the computation and provide a graph expansion-based characterization to succinctly capture the computation structure, providing new insights into the problem of distributed computation of arbitrary functions. Full article

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

The aim of this paper is to discuss both higher-order asymptotic expansions and skewed approximations for the Bayesian discrepancy measure used in testing precise statistical hypotheses. In particular, we derive results on third-order asymptotic approximations and skewed approximations for univariate posterior distributions, including cases with nuisance parameters, demonstrating improved accuracy in capturing posterior shape with little additional computational cost over simple first-order approximations. For third-order approximations, connections to frequentist inference via matching priors are highlighted. Moreover, the definition of the Bayesian discrepancy measure and the proposed methodology are extended to the multivariate setting, employing tractable skew-normal posterior approximations obtained via derivative matching at the mode. Accurate multivariate approximations for the Bayesian discrepancy measure are then derived by defining credible regions based on an optimal transport map that transforms the skew-normal approximation to a standard multivariate normal distribution. The performance and practical benefits of these higher-order and skewed approximations are illustrated through two examples. Full article

Aimed at dealing with the problems of high reliability solution cost and low solution accuracy under random excitation, especially Gaussian white noise excitation, this paper proposes a probability density evolution and reliability analysis method for nonlinear gear transmission systems under Gaussian white noise excitation based on the path integration method. This method constructs an efficient probability density evolution framework by combining the path integration method, the Chapman–Kolmogorov equation, and the Laplace asymptotic expansion method. Based on Rice’s theory and combined with the adaptive Gauss–Legendre integration method, the transient and cumulative reliability of the system are path integration method calculated. The research results show that in the periodic response state, Gaussian white noise leads to the diffusion of probability density and peak attenuation, and the system reliability presents a two-stage attenuation characteristic. In the chaotic response state, the intrinsic dynamic instability of the system dominates the evolution of the probability density, and the reliability decreases more sharply. Verified by Monte Carlo simulation, the method proposed in this paper significantly outperforms the traditional methods in both computational efficiency and accuracy. The research reveals the coupling effect of Gaussian white noise random excitation and nonlinear dynamics, clarifies the differences in failure mechanisms of gear systems in periodic and chaotic states, and provides a theoretical basis for the dynamic reliability design and life prediction of nonlinear gear transmission systems. Full article

For a massive scalar field in a fixed Schwarzschild background, the radial wave equation obeyed by Fourier modes is first studied. After reducing such a radial wave equation to its normal form, we first study approximate solutions in the neighborhood of the origin, horizon and point at infinity, and then we relate the radial with the Heun equation, obtaining local solutions at the regular singular points. Moreover, we obtain the full asymptotic expansion of the local solution in the neighborhood of the irregular singular point at infinity. We also obtain and study the associated integral representation of the massive scalar field. Eventually, the technique developed for the irregular singular point is applied to the homogeneous equation associated with the inhomogeneous Zerilli equation for gravitational perturbations in a Schwarzschild background. Full article

This article describes an effective computing method for singularly perturbed parabolic problems with small negative shifts in convection and reaction terms. To handle the small negative shifts, the Taylor series expansion is used. The asymptotically equivalent singularly perturbed parabolic convection–diffusion–reaction problem is then discretized with the Crank–Nicolson method on a uniform mesh for the time derivative and a hybrid method on Shishkin-type meshes for the space derivative. The method’s stability and parameter-uniform convergence are established. To substantiate the theoretical findings, the numerical results are presented in tables and graphs are plotted. The present results improve the existing methods in the literature. Due to the effect of the small negative shifts in Examples 1 and 2, the numerical results using Shishkin and Bakhvalov–Shishkin meshes are almost the same. Since there are no small shifts in Examples 3 and 4, the numerical results using the Bakhvalov–Shishkin mesh are more efficient than using the Shishkin mesh. We conclude that the present method using the Bakhvalov–Shishkin mesh performs well for singularly perturbed problems without small negative shifts. Full article

We reformulate holographic space–time models in terms of Hilbert bundles over the space of the time-like geodesics in a Lorentzian manifold. This reformulation resolves the issue of the action of non-compact isometry groups on finite-dimensional Hilbert spaces. Following Jacobson, I view the background geometry as a hydrodynamic flow, whose connection to an underlying quantum system follows from the Bekenstein–Hawking relation between area and entropy, generalized to arbitrary causal diamonds. The time-like geodesics are equivalent to the nested sequences of causal diamonds, and the area of the holoscreen (The holoscreen is the maximal $d-2$ volume (“area”) leaf of a null foliation of the diamond boundary. I use the term area to refer to its volume.) encodes the entropy of a certain density matrix on a finite-dimensional Hilbert space. I review arguments that the modular Hamiltonian of a diamond is a cutoff version of the Virasoro generator $L_0$ of a $1+1$-dimensional CFT of a large central charge, living on an interval in the longitudinal coordinate on the diamond boundary. The cutoff is chosen so that the von Neumann entropy is $\ln {\mathrm{D}}_{\diamond },$ up to subleading corrections, in the limit of a large-dimension diamond Hilbert space. I also connect those arguments to the derivation of the ’t Hooft commutation relations for horizon fluctuations. I present a tentative connection between the ’t Hooft relations and $U\left(1\right)$ currents in the CFTs on the past and future diamond boundaries. The ’t Hooft relations are related to the Schwinger term in the commutator of the vector and axial currents. The paper in can be read as evidence that the near-horizon dynamics for causal diamonds much larger than the Planck scale is equivalent to a topological field theory of the ’t Hooft CR plus small fluctuations in the transverse geometry. Connes’ demonstration that the Riemannian geometry is encoded in the Dirac operator leads one to a completely finite theory of transverse geometry fluctuations, in which the variables are fermionic generators of a superalgebra, which are the expansion coefficients of the sections of the spinor bundle in Dirac eigenfunctions. A finite cutoff on the Dirac spectrum gives rise to the area law for entropy and makes the geometry both “fuzzy” and quantum. Following the analysis of Carlip and Solodukhin, I model the expansion coefficients as two-dimensional fermionic fields. I argue that the local excitations in the interior of a diamond are constrained states where the spinor variables vanish in the regions of small area on the holoscreen. This leads to an argument that the quantum gravity in asymptotically flat space must be exactly supersymmetric. Full article

In this paper, we consider the Neumann elliptic problem $\left({\mathcal{I}}_{\epsilon }\right)$: $-\Delta u+V\left(x\right)u=u^{p-\epsilon }$, $u>0$ in $\mathsf{\Omega }$, $\partial u/\partial \nu =0$ on $\partial \mathsf{\Omega }$, where $\mathsf{\Omega }$ is a bounded smooth domain in ${\mathbb{R}}^n$, $n\ge 3$, $p+1=2n/(n-2)$ is the critical Sobolev exponent, $\epsilon$ is a small positive parameter and V is a smooth positive function on $\overline{\mathsf{\Omega }}$. First, in contrast to the case where solutions converge weakly to zero, we rule out the existence of interior bubbling solutions with a non-zero weak limit in small-dimensional domains. Second, for large-dimensional domains, we construct both simple and non-simple interior bubbling solutions with residual mass. This construction allows us to establish the multiplicity results. The proofs of these results are based on refined asymptotic expansions of the gradient of the associated Euler–Lagrange functional. Full article

The production of advanced therapy medicinal products (ATMP) is a long and highly technical process, resulting in a high cost per dose, which reduces the number of eligible patients. There is a critical need for a closed and sample-free monitoring system to perform the numerous quality controls required. Current monitoring methods are not optimal, mainly because they require the system to be opened up for sampling and result in material losses. White light spectroscopy has emerged as a technique for sample-free control compatible with closed systems. We have recently proposed its use to monitor cultures of CEM-C1 cell lines. In this paper, we apply this method to T-cells isolated from healthy donor blood samples. The main differences between cell lines and human primary T-cells lie in the slightly different shape of their absorption spectra and in the dynamics of cell expansion. T-cells do not multiply exponentially, resulting in a non-constant generation time. Cell expansion is described by a power-law model, which allows for the definition of instantaneous generation times. A correlation between the linear asymptotic behavior of these generation times and the initial cell concentration leads to the hypothesis that this could be an early predictive marker of the final culture concentration. To the best of our knowledge, this is the first time that such concepts have been proposed. Full article

This article investigates the boundary value problem for an extended stationary system of Nernst–Planck–Poisson equations, corresponding to a mathematical model of the influence of changes in the equilibrium coefficient on the transport of ions of a binary salt in the diffusion layer. Dimensionless variables were introduced using characteristic parameter values. As a result, a dimensionless boundary value problem was obtained, which is singularly perturbed, containing a small parameter in the derivative of the Poisson equation and, additionally, another regular small parameter. A similarity theory was developed: trivial and non-trivial similarity criteria and their physical meaning were determined, which allowed for the identification of general properties of the solutions. A numerical investigation of the boundary value problem was conducted using the finite element method. With an increase in the initial solution concentration, the value of the small parameter entering singularly decreases, reaching values on the order of 10⁻¹² and below, leading to computational difficulties that prevent a comprehensive analysis of the influence of changes in the equilibrium coefficient on salt ion transport. In this regard, an analytical solution to the problem was constructed, based on dividing the solution domain into several subdomains (regions of electroneutrality, extended space charge region, quasi-equilibrium region, recombination region, intermediate layer), in each of which the problem is solved differently, followed by matching these solutions. Verification of the analytical solution was carried out by comparing it with the numerical solution. The advantage of the obtained analytical solution is the possibility of a comprehensive analysis of the influence of the dissociation/recombination reaction of water molecules on salt ion transport over a wide range of real changes in the concentration and composition of the electrolyte solution and other input parameters. This boundary value problem serves as a benchmark for constructing asymptotic solutions for other singularly perturbed boundary value problems in membrane electrochemistry. Full article

With the rapid expansion of the internet and accelerated information dissemination, rumors pose a significant threat to social stability. Effective rumor control requires a thorough understanding of propagation mechanisms. This study develops a Caputo fractional-order IDSR rumor propagation model with time delays. The equilibrium points are identified, and the local asymptotic stability of the system at the positive equilibrium is analyzed. Additionally, the conditions for Hopf bifurcation and its impact on the rumor dynamics are examined. Numerical simulations validate the theoretical findings. Full article