Rigorous Derivation Research Articles

Semiclassical instanton theory for reaction rates at any temperature: How a rigorous real-time derivation solves the crossover temperature problem

Instanton theory relates the rate constant for tunneling through a barrier to the periodic classical trajectory on the upturned potential energy surface, whose period is τ = ℏ/(kBT). Unfortunately, the standard theory is only applicable below the “crossover temperature,” where the periodic orbit first appears. This paper presents a rigorous semiclassical (ℏ → 0) theory for the rate that is valid at any temperature. The theory is derived by combining Bleistein’s method for generating uniform asymptotic expansions with a real-time modification of Richardson’s flux-correlation function derivation of instanton theory. The resulting theory smoothly connects the instanton result at low temperature to the parabolic correction to Eyring transition state theory at high-temperature. Although the derivation involves real time, the final theory only involves imaginary-time (thermal) properties, consistent with the standard version of instanton theory. Therefore, it is no more difficult to compute than the standard theory. The theory is illustrated with application to model systems, where it is shown to give excellent numerical results. Finally, the first-principles approach taken here results in a number of advantages over previous attempts to extend the imaginary free-energy formulation of instanton theory. In addition to producing a theory that is a smooth (continuously differentiable) function of temperature, the derivation also naturally incorporates hyperasymptotic (i.e., multi-orbit) terms and provides a framework for further extensions of the theory.

Bell vs. Bell: A Ding-Dong Battle over Quantum Incompleteness

Does determinism (or even the incompleteness of quantum mechanics) follow from locality and perfect correlations? In a 1964 paper, John Bell gave the first demonstration that quantum mechanics is incompatible with local hidden variables. Since then, a vigorous debate has rung out over whether he relied on an assumption of determinism or instead, as he later claimed in a 1981 paper, derived determinism from assumptions of locality and perfect correlation. This paper aims to bring clarity to the debate via simple examples and rigorous results. It is first recalled, via quantum and classical counterexamples, that the weakest statistical form of locality consistent with Bell’s 1964 paper (parameter independence) is insufficient for the derivation of determinism. Attention is then turned to critically assess Bell’s appeal to the Einstein–Rosen–Podolsky (EPR) incompleteness argument to support his claim. It is shown that this argument is itself incomplete, via counterexamples that expose two logical gaps. Closing these gaps via a strong “counterfactual” reality criterion enables a rigorous derivation of both determinism and parameter independence, and in this sense justifies Bell’s claim. Conversely, however, it is noted that whereas the EPR argument requires a weaker “measurement choice” assumption than Bell’s demonstration, it nevertheless leads to a similar incompatibility with quantum predictions rather than quantum incompleteness.

Euler’s Formula and Its Applications in Modern Mathematics

This article examines the various uses of Euler‘s formula in complex analysis, topology, number theory, and other mathematical fields. Renowned for its mathematical elegance, exponential and trigonometric functions are closely related according to Euler‘s formula. acting as a fundamental basis for various key developments in these fields. By conducting a thorough analysis, the study reveals the extensive significance and lasting impact of Euler’s formula in both theoretical and applied mathematics. The research employs a comprehensive literature review, enhanced by rigorous mathematical derivations and practical examples, to demonstrate the widespread applicability and versatility of Euler’s formula in a wide range of contexts. The findings underscore that Euler’s formula not only holds a central position in theoretical mathematics but also plays a crucial role in engineering, quantum mechanics, signal processing, and physics. This study contributes to a deeper understanding of the intrinsic relationships within mathematical formulas, emphasizing their far-reaching practical relevance and potential for future research and technological innovation.

Dynamic modeling method for constrained system with singular mass matrices

The dynamic model is beneficial for system control design, especially when it is related to precise force adjustment. Traditional modeling methods make it difficult to address multi-body systems with singular mass matrices or are computationally expensive. In this paper, an approach termed the Extended Rosenberg Embedding Method for dynamic modeling is presented. By incorporating the constraints directly into the Fundamental Equation, the proposed approach enables the description of the system motion in two separate equations, which can reduce the computational cost of the constrained dynamic model. This method provides a new way to establish motion equations, regardless of whether the system is subject to holonomic or non-holonomic constraints. Moreover, as the method does not impose direct requirements on the rank of the mass matrix, it is capable of handling the modeling of multi-body systems with singular mass matrices. The validity of the proposed method is substantiated through rigorous mathematical derivation, while its accuracy and computational efficiency are corroborated through the examination of two numerical examples.

Interfacial dynamic impermeable crack analysis in dissimilar piezoelectric materials by a new interaction integral

Dynamic intensity factors (IFs) provide an important parameter for assessing the failure risk of an interfacial crack for piezoelectric composites exposed to electromechanical impact loadings. In the present research, a new dynamic interaction integral (I-integral) is built for computing the dynamic stress IFs (SIFs) and electric displacement IF (EDIF) of an interfacial crack located in dissimilar inhomogeneous piezoelectric media. Through proper selection of auxiliary variables, the domain expression of I-integral for the inhomogeneous piezoelectric bi-materials does not need to take into account any derivative term for any piezoelectric material property. Further, after rigorous theoretical derivation, the resulting I-integral is still valid for complex models where the integration domain contains other multiple interfaces, regardless of whether the extra materials are straight or curved. Incorporating the modified extended finite element approach, the accuracy is confirmed by checking the dynamic IFs extracted from the I-integral with referenced results. The domain-independence is tested by varying ranges of integration domains for inhomogeneous piezoelectric bi-materials and multi-interface piezoelectric composites. Finally, typical examples are employed to discuss the influences of the direction and magnitude of electric impact loading, combination of polarizations, inhomogeneous degree of piezoelectric bi-materials and complex distribution of diverse material properties on the dynamic IFs.

Vacancy formation free energy in concentrated alloys: Equilibrium vs. random sampling

Special Quasi-random Structures (SQSs) are often used to model disordered alloys in small simulation cells. Yet, SQS-based sampling yields defect formation energies that do not match the equilibrium values, for instance measured in atomic Monte Carlo simulations, due to the lack of chemical short-range order in random samples. In this paper, our approach to computing chemical potentials in alloys through random sampling techniques is extended to the computation of vacancy formation free energies. By rigorous thermodynamic derivation, we show that vacancy formation free energies computed from randomly sampled local chemical environments must be corrected to ensure its consistency between various available calculations. Indeed, irrespective of the choice of the species being replaced by a vacancy, we should arrive at the same value of the vacancy formation free energy. We propose a simple way to compute this correction term, and compare the outcome with equilibrium Monte Carlo results and with approximations found in the literature.

Kac–Ward Solution of the 2D Classical and 1D Quantum Ising Models

AbstractWe give a rigorous derivation of the free energy of (i) the classical Ising model on the triangular lattice with translation-invariant coupling constants and (ii) the one-dimensional quantum Ising model. We use the method of Kac and Ward. The novel aspect is that the coupling constants may have negative signs. We describe the logarithmic singularity of the specific heat of the classical model and the validity of the Cimasoni–Duminil-Copin–Li formula for the critical temperature. We also discuss the quantum phase transition of the quantum model.

Rigorous Derivation of Michaelis–Menten Kinetics in the Presence of Slow Diffusion

Rigorous Derivation of Michaelis–Menten Kinetics in the Presence of Slow Diffusion

Uncertainty analysis method of high-speed wind tunnel standard model test results based on GUM and MCM

Abstract The standard model is a standard calibration model that has a well-known geometric shape and can represent the aerodynamic layout characteristics of a certain type of aircraft. It has many functions, including evaluating the accuracy of wind tunnel test data, verifying the integrity of the wind tunnel flow field and measurement system, and developing and verifying wind tunnel test technology. Conducting wind tunnel tests using standard models is a key step in realizing the functions of standard models. However, all existing literature on standard model tests has not effectively evaluated the uncertainty of the test results, which makes it impossible to evaluate the reliability of the test results. Based on the GUM and MCM uncertainty analysis methods, this paper establishes a method for quantitatively evaluating the uncertainty of the test results of high-speed wind tunnel standard model tests through rigorous mathematical derivation, providing technical support for precisely quantifying the uncertainty of standard model test results.

A novel high‐isolation dual‐polarized air‐patch antenna based on composite decoupling structure

AbstractThis paper presents a novel wideband dual‐polarized air‐patch antenna with enhanced port isolation. The proposed design consists of a suspended metal patch, two feeding slots, and novel composite decoupling structures (CDS). The CDS is composed of two reactive loads (RLs), an additional square slot (ASS), and two neutralizing lines (NLs). The reactive loads and one NL are connected to the feeding slots. The other NL is connected to the ASS. The CDS plays a critical role in achieving wideband high port isolation. The decoupling principles are illustrated by using a circuit model and through rigorous mathematical derivation. Design evolution of the proposed antenna (Ant 1–5) was carefully explained. Experimental results demonstrate that the proposed antenna has a −15 dB refection coefficient bandwidth covering the frequency range of 3.32–3.67 GHz. Additionally, it exhibits high port isolation (>35 dB), low cross polarization, high gain, and excellent total efficiency (>85%).

Explicit and approximate solutions for a classical hyperbolic fragmentation equation using a hybrid projected differential transform method

Population balance equations are widely used to study the evolution of aerosols, colloids, liquid–liquid dispersion, raindrop fragmentation, and pharmaceutical granulation. However, these equations are difficult to solve due to the complexity of the kernel structures and initial conditions. The hyperbolic fragmentation equation, in particular, is further complicated by the inclusion of double integrals. These challenges hinder the analytical solutions of number density functions for basic kernel classes with exponential initial distributions. To address these issues, this study introduces a new approach combining the projected differential transform method with Laplace transform and Padé approximants to solve the hyperbolic fragmentation equation. This method aims to provide accurate and efficient explicit solutions to this challenging problem. The approach's applicability is demonstrated through rigorous mathematical derivation and convergence analysis using the Banach contraction principle. Additionally, several numerical examples illustrate the accuracy and robustness of this new method. For the first time, new analytical solutions for number density functions are presented for various fragmentation kernels with gamma and other initial distributions. This method significantly enhances solution quality over extended periods using fewer terms in the truncated series. The solutions are compared and verified against the finite volume method and the homotopy perturbation method, showing that the coupled approach not only estimates number density functions accurately but also captures integral moments with high precision. This research advances computational methods for particle breakage phenomena, offering potential applications in various industrial processes and scientific disciplines.

Determination of the discreteness correction in dislocation equations for sphalerite crystals

Abstract The discreteness correction for simple lattices has been determined, however, there is still no detailed and rigorous derivation of the discreteness correction for compound lattices due to the relatively complex structure (simple lattices have only one set per slip system, while complex lattices have at least two sets). A lattice dynamics model which takes into account the energy increments caused by changes in bond length and bond angle is constructed for sphalerite crystals and the relationships between discreteness parameters and elastic constants are determined for the slip system. Compared to glide set, the discreteness correction for shuffle set is much larger due to it is mainly contributed by the interactions between nearest neighbor atoms. Based on the results obtained from the model, the dislocations in ZnS crystal are investigated. The theoretical prediction result of Peierls stress for shuffle dislocation closely matches the experimental critical shear stress. It is inferred that the initial plastic deformation of ZnS is closely related to the movement of shuffle dislocations.

Resilient robust model predictive load frequency control for smart grids with air conditioning loads

AbstractThis paper investigates the robust model predictive load frequency control problem for smart grids with wind power under cyber attacks. To accommodate intermittent power generation, the demand response of the system is considered by involving the air conditioning loads in the frequency regulation. In addition, the system uncertainties produced by the air conditioning load users and wind turbines when replacing traditional generator sets are considered. By using the cone complementary linearization algorithm and the linear matrix inequality technique, a resilience robust model predictive control strategy with mixed performance indexes is proposed. Furthermore, a rigorous derivation of the recursive feasibility of robust model predictive control is given. Finally, the simulation results of the two‐area load frequency control scheme show that the proposed model predictive control strategy is capable of realizing the load frequency control of the multi‐area smart grid and is robust to the parameter uncertainties and frequency regulation of the system. The results also show that the proposed model predictive control strategy has some resistance to cyber attacks and external disturbances.

An impartial framework to investigate demosaicking input embedding options

Convolutional Neural Networks (CNNs) have proven highly effective for demosaicking, transforming raw Color Filter Array (CFA) sensor samples into standard RGB images. Directly applying convolution to the CFA tensor can lead to misinterpretation of the color context, so existing demosaicking networks typically embed the CFA tensor into the Euclidean space before convolution. The most prevalent embedding options are Reordering and Pre-interpolation. However, it remains unclear which option is more advantageous for demosaicking. Moreover, no existing demosaicking network is suitable for conducting a fair comparison. As a result, in practice, the selection of these two embedding options is often based on intuition and heuristic approaches. This paper addresses the non-comparability between the two options and investigates whether pre-interpolation contributes additional knowledge to the demosaicking network. Based on rigorous mathematical derivation, we design pairs of end-to-end fully convolutional evaluation networks, ensuring that the performance difference between each pair of networks can be solely attributed to their differing CFA embedding strategies. Under strictly fair comparison conditions, we measure the performance contrast between the two embedding options across various scenarios. Our comprehensive evaluation reveals that the prior knowledge introduced by pre-interpolation benefits lightweight models. Additionally, pre-interpolation enhances the robustness to imaging artifacts for larger models. Our findings offer practical guidelines for designing imaging software or Image Signal Processors (ISPs) for RGB cameras.

Quantum logarithmic multifractality

Through a combination of rigorous analytical derivations and extensive numerical simulations, this work reports an exotic multifractal behavior, dubbed “logarithmic multifractality,” in effectively infinite-dimensional systems undergoing the Anderson transition. In contrast to conventional multifractality observed in finite dimensions, logarithmic multifractality at infinite dimension introduces an algebraic behavior with respect to the logarithm of system size or time. We demonstrate this phenomenon across eigenstate statistics, spatial correlations, and wave packet dynamics. Our findings offer crucial insights into strong finite-size effects and slow dynamics in complex systems undergoing the Anderson transition, such as the many-body localization transition. Published by the American Physical Society 2024

A counterfactual explanation method based on modified group influence function for recommendation

In recent years, recommendation explanation methods have received widespread attention due to their potentials to enhance user experience and streamline transactions. In scenarios where auxiliary information such as text and attributes are lacking, counterfactual explanation has emerged as a crucial technique for explaining recommendations. However, existing counterfactual explanation methods encounter two primary challenges. First, a substantial bias indeed exists in the calculation of the group impact function, leading to the inaccurate predictions as the counterfactual explanation group expands. In addition, the importance of collaborative filtering as a counterfactual explanation is overlooked, which results in lengthy, narrow, and inaccurate explanations. To address such issues, we propose a counterfactual explanation method based on Modified Group Influence Function for recommendation. In particular, via a rigorous formula derivation, we demonstrate that a simple summation of individual influence functions cannot reflect the group impact in recommendations. After that, building upon the improved influence function, we construct the counterfactual groups by iteratively incorporating the individuals from the training samples, which possess the greatest influence on the recommended results, and continuously adjusting the parameters to ensure accuracy. Finally, we expand the scope of searching for counterfactual groups by incorporating the collaborative filtering information from different users. To evaluate the effectiveness of our method, we employ it to explain the recommendations generated by two common recommendation models, i.e., Matrix Factorization and Neural Collaborative Filtering, on two publicly available datasets. The evaluation of the proposed counterfactual explanation method showcases its superior performance in providing counterfactual explanations. In the most significant case, our proposed method achieves a 17% lead in terms of Counterfactual precision compared to the best baseline explanation method.

Analytic Modelling of 2-D Transient Heat Conduction with Heat Source Under Mixed Boundary Constraints by Symplectic Superposition

Abstract Many studies have been conducted on 2-D transient heat conduction, but analytic modelling is still uncommon for the cases with complex boundary constraints due to the mathematical challenge. With an unusual symplectic superposition method, this paper reports new analytic solutions to 2-D isotropic transient heat conduction problems with heat source over a rectangular region under mixed boundary constraints at an edge. With the Laplace transform, the Hamiltonian governing equation is derived. The applicable mathematical treatments, e.g., the variable separation and the symplectic eigenvector expansion in the symplectic space, are implemented for the fundamental solutions whose superposition yields the ultimate solutions. Benchmark results obtained by the present method are tabulated, with verification by the finite element solutions. Instead of the conventional Euclidean space, the present symplectic-space solution framework has the superiority on rigorous derivations without pre-determining solution forms, which may be extended to more issues with the complexity caused by mixed boundary constraints.

Incompressible Navier–Stokes limit from nonlinear Vlasov–Fokker–Planck equation

The aim of this paper is to justify the rigorous derivation of the incompressible Navier–Stokes equations from the nonlinear Vlasov–Fokker–Planck (VFP) equation with a constant temperature. Under the incompressible Navier–Stokes scaling, we first establish the global existence of regular solutions to the rescaled nonlinear VFP equation near the Maxwellian, obtaining some uniform bound estimates. We then show the strong convergence of solution to the nonlinear VFP equation towards the incompressible Navier–Stokes system.

Robust internal model control based on a novel generalized extended state observer and its application on a two-inertia system

Disturbance observer (DOB) and extended state observer (ESO) are extensively utilized to handle external disturbances and model uncertainties in the control system. Nevertheless, the integration of these two methods to improve disturbance suppression remains an open question. In this research, the disturbance compensation mechanism of DOB is employed to compensate the disturbance estimation error of ESO, thereby achieving an effective integration of DOB and ESO. Additionally, a generalized ESO (GESO) is proposed to replace ESO. A robust internal mode control (RIMC) scheme is then developed by incorporating GESO into a two-degree-of-freedom internal mode control (TDF-IMC) framework. Moreover, the equivalence of RIMC and classical TDF-IMC is given by a rigorous derivation under the frequency domain description. Finally, the RIMC is applied to the control of a two-inertia system to verify its superiority in terms of robustness, disturbance rejection, and resonance suppression.

Analysis of signal-to-noise ratio of spatial heterodyne spectroscopy

We have established a novel spatial heterodyne spectroscopy (SHS) signal-to-noise ratio (SNR) model, relating the spectral SNR to the spectral band and resolution, which helps guide the instrument's design and optimization and assess the spectral quality. The experimental and simulation results show that the spectral resolution affects the SNR differently under different noise types and spectral characteristics of the measured targets, which fully validates the novel SHS SNR model. Despite the disadvantage of multiplexing in SHS, we determine the SNR advantage of SHS over the grating spectroscopy (GS) through rigorous theoretical derivations. In 94.34 % of the polychromatic light detections, the average SNR of SHS is 2–31 times higher than that of GS. In emission spectra detections, the SNR gain of SHS relative to GS is up to 1–2 orders of magnitude. In additive noise domination, the SNR gain reaches 1–3 orders of magnitude.