Rigorous Derivation Research Articles

Nonlinear RNN with noise-immune: A robust and learning-free method for hyperspectral image target detection

While the recurrent neural network (RNN) has achieved remarkable performance on dynamic and control tasks, its applications to image processing, particularly target detection are limited. Challenges arise from differences between the two domains, such as the way for merging time information into static problems and variances of dynamic and static solving methods. To this end, we first extend the existing constrained energy minimization (CEM)-based detection scheme to a dynamic version, e.g. dynamic reinforced CEM (DRCEM), which injects the dynamic information. After that, aided by the rigorous mathematical derivation and optimization theory, the DRCEM is merged into the RNN solution framework. To enhance the robustness and convergence of the existing RNN solutions for improving DRCEM performance, the nonlinear and bounded-constraint RNN (NBCRNN) is designed by developing a novel nonlinear activation function, then applying the proposed model to implement the DRCEM scheme. The corresponding theorem results reveal the proposed model possesses global convergence and enhanced robustness. Compared to state-of-the-art works, the DRCEM solved by the NBCRNN model detection method achieves better detection accuracy, with 1.82% improvement in terms of the Kappa coefficient, and reduces the residual error from 10−4 to 10−7. Furthermore, our detection method is able to preserve the detection accuracy in presence of noise perturbated. To the best of our knowledge, it is the first work to develop the zeroing-type RNN for hyperspectral image target detection. The code and models are publicly available at Github DRCEM_NBCRNN Code Implementation.

Arctic curves of the four-vertex model

We consider the four-vertex model with a special choice of fixed boundary conditions giving rise to limit shape phenomena. More generally, the considered boundary conditions relate vertex models to scalar products of off-shell Bethe states, boxed plane partitions, and fishnet diagrams in quantum field theory. In the scaling limit, the model exhibits the emergence of an arctic curve separating a central disordered region from six frozen ‘corners’ of ferroelectric or anti-ferroelectric type. We determine the analytic expression of the interface by means of the Tangent Method. We supplement this heuristic method with an alternative, rigorous derivation of the arctic curve. This is based on the exact evaluation of suitable correlation functions, devised to detect spatial transition from order to disorder, in terms of the partition function of some discrete log-gas associated to the orthogonalizing measure of the Hahn polynomials. As a by-product, we also deduce that the arctic curve’s fluctuations are governed by the Tracy–Widom distribution.

Refined beam finite element model for thin-walled multi-cell box girders considering distortion and secondary distortional moment deformation effect

In this paper, a novel one-dimensional refined beam finite element model (FEM) accounting for non-uniform distortional warping and secondary distortional moment deformation effect (SDMDE) is formulated for the distortional analysis of thin-walled multi-cell box girders with cantilevered flanges. The extra distortional angle is introduced to incorporate SDMDE, and the validity of which is proved through rigorous theoretical derivation. The in-plane and out-plane deformations at any point around the edge of transverse section built upon the distortional center are unified in the same form and described in detail. The governing differential equations in terms of two generalized displacements, distortional angle χ and primary distortional angle curvature Θ, are derived based on the principle of minimum potential energy (MPE) and solved exactly so as to provide non-polynomial shape functions for the establishment of element stiffness matrix and nodal load vector. A series of multi-cell rectangular hollow section (RHS) girders are analyzed and particular extensions are given to single-cell and multi-cell bridge girders with cantilevered flanges. Comparisons to the results generated from corresponding ones provided by pioneering work or shell FEM analyses not only validate the correctness and reliability of the present beam element formulations but also highlight the significance of the inclusion of SDMDE when the cantilevered flanges are arranged.

Analysis on the collapse stress of auxetic tubular anti-tetrachiral structures

The growing demands for enhanced impact resistance in modern engineering necessitate the exploration of novel materials with superior properties. Among these materials, the tubular anti-tetrachiral structure (TATS) emerges as a promising 3D metamaterial, exhibiting exceptional characteristics owing to its auxetic effect. Nevertheless, the investigation into the collapse stress of TATSs subjected to impact remains limited. Thus, this study aims to examine the deformation modes exhibited by TATSs under both low and high velocity impacts, while also deriving the theoretical expression for collapse stress through rigorous theoretical derivations. Furthermore, we delve into the analysis of the influence exerted by impact velocity and structural geometric parameters on the collapse stress of the TATS. Sensitivity analysis is performed to identify structural geometric parameters that display heightened sensitivity, and the interplay between impact velocity and these parameters is carefully scrutinized. The findings presented in this paper furnish a robust theoretical foundation for the design and exploration of TATSs.

From BGK-alignment model to the pressured Euler-alignment system with singular communication weights

This paper is devoted to a rigorous derivation of the isentropic Euler-alignment system with singular communication weights ϕα(x)=|x|−α for some α>0. We consider a kinetic BGK-alignment model consisting of a kinetic BGK-type equation with a singular Cucker-Smale alignment force. By taking into account a small relaxation parameter, which corresponds to the asymptotic regime of a strong effect from the BGK operator, we quantitatively derive the isentropic Euler-alignment system with pressure p(ρ)=ργ, γ=1+2d from that kinetic equation.

Underlying mathematical relations for efficient COMET response expansion coefficient and core calculations

The coarse mesh transport code COMET, a hybrid stochastic-deterministic neutronics solver with formidable computation speed, can provide high fidelity transport solutions to various reactor core problems. In this paper, to take advantage of local rotational and reflection symmetry existing in many reactor cores (e.g., fuel lattices and reflector blocks), the underlying mathematical relations among the response expansion coefficients of symmetric coarse meshes are developed to further improve the computational efficiency in both COMET response expansion coefficient generation and core calculation. This is done by a rigorous derivation of the transformation matrices for the angular and spatial expansion moments resulting from a rotation of a coarse mesh by an arbitrary angle or reflection of a coarse mesh. As a result, the relations for the response expansion coefficients of a coarse mesh with rotational symmetry can be then written as the Kronecker product of those transformation matrices. For coarse meshes with reflection symmetry, the sparse property, the outgoing surface symmetry property and the incident surface symmetry property of response expansion coefficients are rigorously derived. These underlying mathematical relations are implemented into COMET and evaluated on three stylized 3D PWR whole core benchmark problems. The COMET results using the response expansion coefficient library based on the underlying mathematical relations were compared to those using the library directly generated by Monte Carlo for all surfaces. It was found that the eigenvalues as well as the pin and assembly fission densities using the two libraries are in statistical agreement as expected. This indicates that the new COMET using the underlying mathematical relations maintains the high fidelity of the original COMET while improving computational efficiency by 10.7 and 3.7 times for COMET response expansion coefficient generation and core calculation, respectively. This also reduces the size of the library by 10.7 times.

Novel Compact Polarized Martian Wind Imaging Interferometer

The Mars Atmospheric Wind Imaging Interferometer offers several advantages, notably its high throughput, enabling the acquisition of precise and high vertical resolution data on the temperature and wind fields in the Martian atmosphere. Considering the current absence of such an Interferometer, this paper introduces a novel Mars wind field imaging interferometer. In analyzing the photochemical model of O2 (a1Δg) 1.27 μm molecular airglow radiation in the Martian atmosphere and considering the impact of instrument signal-to-noise ratio (SNR), we have chosen an optical path difference (OPD) of 8.6 cm for the interferometer. The all-solid-state polarized wind imaging interferometer is miniaturized by incorporating two arm glasses as the compensation medium in its construction, achieving the effects of field-widening and temperature compensation. Additionally, an F-P Etalon is designed to selectively filter the desired three spectral lines of O2 dayglow, and its effect is evaluated through simulations. The accuracy of the proposed compact Mars polarized wind imaging interferometer for detecting Mars’ wind field and temperature field has been validated through rigorous theoretical derivation and comprehensive computer simulations. The interferometer boasts several advantages, including its compact and small size, static stability, minimal stray light, and absence of moving parts. It establishes the theoretical, technological, and instrumental engineering foundations for future simultaneous static measurement of Martian global atmospheric wind fields, temperature fields, and ozone concentrations from spacecraft, thereby significantly contributing to the dataset for investigating Martian atmospheric dynamics.

Rigorous derivation of the Fick cross-diffusion system from the multi-species Boltzmann equation in the diffusive scaling

We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann equation in a rigorous manner in Sobolev spaces. To this end, we formally show that, in a diffusive scaling, the hydrodynamical limit of the kinetic system is the Fick model supplemented with a closure relation and we give explicit formulae for the macroscopic diffusion coefficients from the Boltzmann collision operator. Then, we provide a perturbative Cauchy theory in Sobolev spaces for the constructed Fick system, which turns out to be a dilated parabolic equation. We finally prove the stability of the system in the Boltzmann equation, ensuring a rigorous derivation between the two models.

Ship-Mounted Cranes Hoisting Underwater Payloads: Transportation Control With Guaranteed Constraints on Overshoots and Swing

In recent years, with the rapid development of marine engineering, ship-mounted crane control for transporting payloads in the water has attracted more attention. Compared with the case of transporting payloads above water, ship-mounted cranes with underwater payloads are more difficult to control. On the one hand, the underwater payload is directly affected by the hydrodynamic force, and the dynamics of ship-mounted cranes is much more complex, nonlinear, and coupled; on the other hand, the unactuated underwater payload swing is quite sensitive to external disturbances; thus, the harsh marine environment will bring great challenges to the anti-swing control of underwater payloads. To address the above issues, this article puts forward a coupling characteristic indicator(CCI)-based nonlinear control method to realize accurate positioning and swing suppression for ship-mounted cranes hoisting payloads in the water, which not only simultaneously suppresses actuated boom overshoots and constrains <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unactuated</i> payload swing, but also indicates whether the coupling terms are beneficial or harmful to make full use of them to improve transient control performance. Rigorous theoretical derivation proves the closed-loop stability. To the best of our knowledge, this is the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">first</i> solution to handle state constraints and utilizing the coupling characteristics of ship-mounted cranes for transferring payloads in the water. Finally, the proposed controller is applied to a self-made hardware platform, and the experimental results show that the designed method achieves satisfactory control performance.

Vibration-Theoretic Approach to Vulnerability Analysis of Nonlinear Vehicle Platoons

This research explores the inherent vulnerability of nonlinear vehicle platoons characterized by the oscillatory behavior triggered by external perturbations. The perturbation exerted on the vehicle platoon is regarded as an external force on an object. Following the mechanical vibration analysis in mechanics, this research proposes a vibration-theoretic approach that advances our understanding of platoon vulnerability from two aspects. First, the proposed approach introduces damping intensity to characterize vehicular platoon vulnerability, which divides platoon oscillations into two types, i.e., underdamped and overdamped. The damping intensity measures the platoon’s recovery strength in responding to perturbations. Second, the proposed approach can obtain the resonance frequency of a nonlinear vehicle platoon, where resonance amplifies platoon oscillation magnitude when the external perturbation frequency equals the platoon’s damping oscillation frequency. The main contribution of this research lies in the analytical derivation of the closed-form formulas of damping intensity and resonance frequency. In particular, the proposed approach formulates platoon dynamics under perturbation as a second-order non-homogeneous ordinary differential equation, enabling rigorous derivations and analyses for platoons with complicated nonlinear car-following behaviors. Through simulations built on real-world data, this paper demonstrates that an overdamped vehicle platoon is more robust against perturbations, and an underdamped platoon can be destabilized easily by exerting a perturbation at the platoon’s resonance frequency. The theoretical derivations and simulation results shed light on the design of reliable platooning control, either for human-driven or automated vehicles, to suppress the adverse effects of oscillations.

On the Emergence of the Deviation from a Poisson Law in Stochastic Mathematical Models for Radiation-Induced DNA Damage: A System Size Expansion.

In this paper, we study the system size expansion of a stochastic model for radiation-induced DNA damage kinetics and repair. In particular, we characterize both the macroscopic deterministic limit and the fluctuation around it. We further show that such fluctuations are Gaussian-distributed. In deriving such results, we provide further insights into the relationship between stochastic and deterministic mathematical models for radiation-induced DNA damage repair. Specifically, we demonstrate how the governing deterministic equations commonly employed in the field arise naturally within the stochastic framework as a macroscopic limit. Additionally, by examining the fluctuations around this macroscopic limit, we uncover deviations from a Poissonian behavior driven by interactions and clustering among DNA damages. Although such behaviors have been empirically observed, our derived results represent the first rigorous derivation that incorporates these deviations from a Poissonian distribution within a mathematical model, eliminating the need for specific ad hoc corrections.

An evaluation model for wall heat transfer effects on micro centrifugal compressors and its application in aerodynamic design

In micro gas turbine generators, the significant temperature difference within close proximity of the hot and cold components induces a substantial heat influx into the micro centrifugal compressor. This heat influx profoundly modifies the aerodynamic characteristics of the compressor. Ensuring a precise evaluation of the performance decline attributable to wall heat transfer effects and systematically integrating these impacts into the aerodynamic design process assumes paramount importance. Previous evaluation models, which involved unnecessary simplifications, suffered from limitations in accuracy. This paper presented a novel evaluation model for the diabatic isentropic efficiency through entropy analysis of the diabatic ideal compression process and the actual compression process. The rigorous theoretical derivation without any simplification empowers this model to faithfully mirror the authentic aerodynamic performance of diabatic compressors. The experimental and numerical verification results indicate that the proposed model is more reliable than the previous ‘preheating-adiabatic compression’ model when evaluating performance degradation stemming from wall heat transfer. The new model exhibits an error of approximately 3.7% at the design point and a mean absolute error of about 6% along the characteristic line of the design speed. Both deviations are deemed acceptable within the purview of engineering assessments. Furthermore, this paper illustrated an instance of the novel model’s application in the aerodynamic design process. A prototype compressor, impeccably designed under adiabatic conditions but falling short of performance benchmarks during the performance test, was redesigned using the proposed method. The experimental results show that through the incorporation of wall heat transfer effects, the redesigned compressor aligns perfectly with design requirements, yielding an efficiency enhancement of roughly 1.5%.

Quantum generalized Calogero–Moser systems from free Hamiltonian reduction

The one-dimensional system of particles with a 1/x2 repulsive potential is known as the Calogero–Moser system. Its classical version can be generalized by substituting the coupling constants with additional degrees of freedom, which span the so(N) or su(N) algebra with respect to Poisson brackets. We present the quantum version of this generalized model. As the classical generalization is obtained by a symplectic reduction of a free system, we present a method of obtaining a quantum system along similar lines. The reduction of a free quantum system results in a Hamiltonian, which preserves the differences in dynamics of the classical system depending on the underlying, orthogonal or unitary, symmetry group. The orthogonal system is known to be less repulsive than the unitary one, and the reduced free quantum Hamiltonian manifests this trait through an additional attractive term ∑i<j−ħ2(xi−xj)2, which is absent when one performs the straightforward canonical Dirac quantization of the considered system. We present a detailed and rigorous derivation of the generalized quantum Calogero–Moser Hamiltonian, we find the spectra and wavefunctions for the number of particles N=2,3, and we diagonalize the Hamiltonian partially for a general value of N.

An analytical method for predicting the soil-structure interaction of axially loaded piles in sands incorporating the two-surface plasticity model

The paper proposed a novel and practical analytical method for predicting the soil-structure interaction in sands. To well capture the stress softening and soil dilatancy behaviors during loading, a developed two-surface plasticity model for interface shearing is adopted to determine the 2-D stress–strain state of the soil-structure interface and the shear band, while the 3-D elastoplastic behavior of soil around the shear band is determined by another developed two-surface plasticity model for triaxial compression. Following a standardized solving procedure, the rigorous derivation for the stress–strain relationship in plastic zone is formulated as a system of first-order differential equations, which are solved as an initial value problem. Combined with the finite difference method (FDM), the load–displacement response of the axially loaded pile and the stress–strain state around the pile are presented. To verify the proposed analytical approach, a FEM simulation is performed using a user-defined subroutine. The soil with different initial void ratios ranging from very dense to loose is also modeled to demonstrate the effect of the initial void ratio on soil-structure response. The proposed analytical approach is also validated against centrifuge tests and field tests, manifesting the validity and capability of the proposed method.

Rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system from a moderately interacting stochastic particle system. Part I Partial differential equation

The aim of this paper is to provide the analysis result for the partial differential equations arising from the rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system from a moderately interacting stochastic particle system. The rigorous derivation is divided into two articles. In this paper, we establish the solution theory of the degenerate parabolic-elliptic Keller-Segel problem and its non-local version, which will be used in the second paper for the discussion of the mean-field limit. A parabolic regularized system is introduced to bridge the stochastic particle model and the degenerate Keller-Segel system. We derive the existence of the solution to this regularized system by constructing approximate solutions, giving uniform estimates and taking the limits, where a crucial step is to obtain the L∞ Bernstein type estimate for the gradient of the approximate solution. Based on this, we obtain the well-posedness of the corresponding non-local equation through perturbation method. Finally, the weak solution of the degenerate Keller-Segel system is obtained by using a nonlinear version of Aubin-Lions lemma.

Event-Triggered Synchronization for Delayed Quaternion-Valued Inertial Fuzzy Neural Networks via Nonreduced Order Approach

This paper mainly studies the synchronization problem for a class of Takagi-Sugeno (T-S) fuzzy quaternion-valued inertial neural networks with time-varying delay through event-triggered control (ETC) scheme. Firstly, a class of quaternion-valued inertial fuzzy neural networks (QVIFNNs) model with time-varying delay is proposed. To avoid the increase of the number of state variables caused by reduced-order method, non-reduced order approach is utilized fully, and a fuzzy exponential gain event-triggered controller is designed so as to occupy the less communication resources. Then, by establishing a novel Lyapunov functional well handling the high-order term, a new synchronization condition is derived under static event-triggered scheme. And then, to further reduce the number of triggers, a dynamic event-triggered condition is developed and the corresponding sufficient criterion is given. Meanwhile, it is proved that the lower bound of event intervals are non-zero positive and the Zeno phenomenons do not exist via rigorous mathematical derivation. Finally, a numerical example is offered to show the effectiveness of the proposed method. A application example for the considered QVIFNNs with time-varying delay is given.

Design of High-Efficiency Dual Band Power Amplifier Based on GaN HEMT

In this paper, a novel dual-frequency impedance converter is proposed, and a rigorous derivation process is given. Based on this, a 0.61/2.6 GHz dual-frequency and high-efficiency power amplifier is designed. The joint simulation results show that the designed power amplifier achieves a maximum power added efficiency (PAE) of 68.3% at 0.61 GHz and achieves a saturated output of 41.53 dBm; At 2.6 GHz, the saturated output power is 40.8b dBm, and the maximum PAE is 56.3%. The designed dual-frequency power amplifier has good working performance.

A Rigorous Derivation of the Functional Renormalisation Group Equation

The functional renormalisation group equation is derived in a mathematically rigorous fashion in a framework suitable for the Osterwalder–Schrader formulation of quantum field theory. To this end, we devise a very general regularisation scheme which retains reflection positivity as well as the infinite degrees of freedom including smoothness. Furthermore, it is shown how the classical limit is altered by the regularisation process leading to an inevitable breaking of translation invariance. We also give precise conditions for the convergence of the obtained theories upon removal of the regularisation.

Rigorous Derivation of the Oberbeck–Boussinesq Approximation Revealing Unexpected Term

We consider a general compressible viscous and heat conducting fluid confined between two parallel plates and heated from the bottom. The time evolution of the fluid is described by the Navier–Stokes–Fourier system considered in the regime of low Mach and Froude numbers suitably interrelated. Surprisingly and differently to the case of Neumann boundary conditions for the temperature, the asymptotic limit is identified as the Oberbeck–Boussinesq system supplemented with non-local boundary conditions for the temperature deviation.

Rigorous analysis of the topologically protected edge states in the quantum spin Hall phase of the armchair ribbon geometry

Studying the edge states of a topological system and extracting their topological properties is of great importance in understanding and characterizing these systems. In this paper, we present a novel analytical approach for obtaining explicit expressions for the edge states in the Kane-Mele model within a ribbon geometry featuring armchair boundaries. Our approach involves a mapping procedure that transforms the system into an extended Su–Schrieffer–Heeger model, specifically a two-leg ladder, in momentum space. Through rigorous derivation, we determine various analytical properties of the edge states, including their wave functions and energy dispersion. Additionally, we investigate the condition for topological transition by solely analyzing the edge states, and we elucidate the underlying reasons for the violation of the bulk-edge correspondence in relatively narrow ribbons. Our findings shed light on the unique characteristics of the edge states in the quantum spin Hall phase of the Kane–Mele model and provide valuable insights into the topological properties of such systems.