Multi-parametric bifurcations of a fractional neural network with multiple delays and inertial terms.

To predict the microstructures of steels, direct coupling of phase-field method and Calculation of Phase Diagrams (CALPHAD) databases is desirable. However, such direct coupling is challenging even for the binary Fe-C system. The first difficulty is that the local equilibrium condition becomes an implicit function, resulting in extremely high computational costs. The second difficulty is that the relationship between phase composition used in phase-field method and site fraction employed in CALPHAD method is nonlinear. In this study, a phase-field model for the Fe-C system is developed by combining Direct CALPHAD Coupling (DCC) model, which explicitly solves the local equilibrium condition, with a chain-rule formulation that links phase composition to site fractions. Numerical tests for δ phase solidification and γ phase peritectic transformation demonstrated that the proposed model satisfies the local equilibrium condition with errors small enough to have no impact on the simulation results and accurately reproduces phase diagram at equilibrium condition. The developed approach provides a framework for simulating microstructure evolution directly coupled with CALPHAD database for steels containing interstitial elements such as carbon and nitrogen.

Cardiac Magnetic Resonance (CMR) imaging is widely used to personalize heart models for cardiac digital twin analysis because of its ability to visualize soft tissues and capture dynamic functions. However, CMR images have an anisotropic nature, characterized by large inter-slice distances and misalignments from cardiac motion. These limitations result in data loss and measurement inaccuracies, hindering the capture of detailed anatomical structures. In this work, we introduce MorphiNet, a novel network that reproduces heart anatomy learned from high-resolution Computed Tomography (CT) images, unpaired with CMR images. MorphiNet encodes the anatomical structure as gradient fields, deforming template meshes into patient-specific geometries. A multilayer graph subdivision network refines these geometries while maintaining dense point correspondence, suitable for downstream computational analysis. MorphiNet achieved the strongest overall trade-off in bi-ventricular myocardium reconstruction on CMR patients with tetralogy of Fallot, with 0.3 higher Dice score and 2.6 lower Hausdorff distance compared to the best existing template-based methods, while achieving comparable geometric accuracy to neural implicit function methods on CT data at 50× faster inference. Cross-dataset validation on the Automated Cardiac Diagnosis Challenge confirmed robust generalization, achieving a 0.7 Dice score with 30% improvement over previous template-based approaches. We validate our anatomical learning approach through the successful restoration of missing cardiac structures and demonstrate significant improvement over standard Loop subdivision. Motion tracking experiments further confirm MorphiNet's capability for cardiac function analysis, including ejection-fraction estimates that correctly identify myocardial dysfunction in tetralogy of Fallot patients. Code and checkpoints are available at https://github.com/MalikTeng/MorphiNetV2.

Recent research has shown that piecewise smooth (PS) functions can be approximated by piecewise linear functions with second order error in the distance to a given reference point. A semismooth Newton type algorithm based on successive application of these piecewise linearizations was subsequently developed for the solution of PS equation systems. In the present work we relax the criterion of local bijectivity of the linearization to local openness. For this purpose a weak implicit function theorem is proved via local mapping degree theory. It is shown that there exist PS functions f : R 2 → R 2 satisfying the weaker criterion where every neighbourhood of the root of f contains a point x such that all elements of the Clarke Jacobian at x are singular. In such neighbourhoods the steps of classical semismooth Newton are not defined, which establishes the new method as an independent algorithm. To further clarify the relation between a PS function and its piecewise linearization, several statements about structure correspondences between the two are proved. Moreover, the influence of the specific representation of the local piecewise linear models on the robustness of our method is studied. An example application from cardiovascular mathematics is given.

Metamaterials can achieve extraordinary properties unattainable in natural materials through sophisticated artificial structural design. This study constructs novel tri-periodic minimal surface configurations based on fundamental structures. By employing surface boundary capture techniques to capture infinitely continuous and smooth surfaces, corresponding fusion functions are derived. New configurations are then established: one fused with rod elements and another featuring smooth fusion of multiple fundamental structures. By integrating the advantages of these configurations, novel metamaterials with both excellent load-bearing and heat transfer properties can be designed. Transition fusion functions are employed to derive implicit function construction methods for each new configuration. Through homogenisation analysis of all novel configurations, this study obtains their mechanical vector data. Combined with the implicit function expressions, this configuration and design methodology establishes a novel theoretical approach for reverse-engineering mechanical metamaterials according to specific requirements.

This paper investigates a predator-prey system with the Allee effect under homogeneous Neumann boundary conditions. Firstly, a priori estimates of the positive steady state solutions are obtained by using the maximum principle, and using the energy method, the nonexistence of nonconstant positive steady states is proved. Secondly, the Turing instability of the positive steady state is discussed, and the existence of nonconstant positive steady state solutions is obtained by the Leray-Schauder degree theory. Then, local and global bifurcations at simple eigenvalues are studied by using the bifurcation theory and the bifurcation direction is determined. Additionally, the local bifurcations at double eigenvalues are analyzed by applying the spatial decomposition and the implicit function theorem. Finally, the analytical results of numerical simulations are verified and supplemented.

In the classic SIR model, infection gives full immunity against any possible reinfection. However, for many important epidemiological situations, immunity is only partial and reinfection is possible. Though these models are mathematically more complex, we are able to find expressions for the epidemic final size. We also generalise these expressions to include vaccination, with a fraction of the population vaccinated before the epidemic, where vaccinees are less susceptible to primary infections than unvaccinated hosts.Partial immunity can be interpreted at the population level as providing either full or no protection to each host, in some proportion (all-or-none immunity). In this scenario, we give analytical expressions (mathematically similar to the SIR final-size) for the cumulative primary infections and the cumulative reinfections in unvaccinated and vaccinated hosts. Alternatively, partial immunity can be interpreted as providing homogeneous imperfect protection to each host (leaky immunity). For this other scenario, we again obtain an implicit equation for the final epidemic size. We break down, in terms of the final size, the number of infections in hosts with or without prior immunity (vaccine- or infection- induced), as well as the number of primary infections and reinfections. Under the leaky immunity assumption, we find a form of reinfection threshold. If the relative host susceptibility to reinfection is above this threshold (which is the inverse of the pathogen's basic reproduction number), transmission rates are high enough to support an endemic disease. Below the reinfection threshold, epidemics are transient. In the all-or-none model, epidemics are always transient.

Abstract In this paper, we study surfaces $$z=\varphi (x,y)$$ z = φ ( x , y ) in Euclidean space that satisfy the equation $$\varphi _{xx}+\varphi _{yy}=\frac{\Lambda }{2}$$ φ xx + φ yy = Λ 2 where $$\Lambda \in \mathbb {R}$$ Λ ∈ R is a real constant. We classify these surfaces when they are the zero level sets of an implicit equation of the type $$f(x)+g(y)+h(z)=0$$ f ( x ) + g ( y ) + h ( z ) = 0 , where f , g and h are smooth functions of one variable. If $$\Lambda =0$$ Λ = 0 , we find a large family of surfaces with interesting symmetry properties. However, if $$\Lambda \not =0$$ Λ ≠ 0 , we show that the surfaces must be either surfaces of revolution or of the type $$z=f(x)+g(y)$$ z = f ( x ) + g ( y ) ; furthermore, explicit parametrizations of these surfaces are obtained.

Polaritons travelling along a hyperbolic medium's surface have recently sparked significant interest in nanophotonics for the unprecedented manipulation ability on light at the nanoscale in a planar way, promising potential nano-optical applications, especially in 2D circuitry. Despite the widespread use of the term "hyperbolic polariton," the hyperbolic nature has predominantly been inferred from numerical solutions of implicit eigenmode equations, rather than established through an explicit analytical isofrequency relation. In this work, we propose an analytical form for describing the iso-frequency contour of the hyperbolic polaritons, showcasing their strictly hyperbolic nature. Such an analytical form is obtained based on the focusing behavior of the hyperbolic polariton and verified by both numerical simulations on commonly used hyperbolic media systems of the hyperbolic polaritons and experimental characterizations on a hyperbolic metamaterial film. The results present a concise and intuitive physical image of its propagation behavior with the angular dispersion, revealing the fundamental mechanism governing the realization of the deeply sub-diffractional volume-confined polariton and provide a groundbreaking methodology in developing novel hyperbolic polaritons-based optical devices.

In this paper, we prove that definable ring topologies on NIP fields are closely connected to NIP integral domains. More precisely, we show that up to elementary equivalence, any NIP topological field arises from an NIP integral domain. As an application, we prove several results about definable ring topologies on NIP fields, including the following. Let [Formula: see text] be an NIP field or expansion of a field. Let [Formula: see text] be a definable ring topology on [Formula: see text]. Then [Formula: see text] is a field topology, and [Formula: see text] is locally bounded. If [Formula: see text] has characteristic [Formula: see text] or finite dp-rank, then [Formula: see text] is “generalized [Formula: see text]-henselian” in the sense of Dittman, Walsberg and Ye, meaning that the implicit function theorem holds for polynomials. If [Formula: see text] has finite dp-rank, then [Formula: see text] must be a topology of “finite breadth” (a [Formula: see text]-topology). Using these techniques, we give some reformulations of the conjecture that NIP local rings are henselian.

High-quality 3D reconstruction of pulmonary segments plays a crucial role in segmentectomy and surgical planning for the treatment of lung cancer. Due to the resolution requirement of the target reconstruction, conventional deep learning-based methods often suffer from computational resource constraints or limited granularity. Conversely, implicit modeling is favored due to its computational efficiency and continuous representation at any resolution. We propose a neural implicit function-based method to learn a 3D surface to achieve anatomy-aware, precise pulmonary segment reconstruction, represented as a shape by deforming a learnable template. Additionally, we introduce two clinically relevant evaluation metrics to comprehensively assess the quality of the reconstruction. Furthermore, to address the lack of publicly available shape datasets for benchmarking reconstruction algorithms, we developed a shape dataset named Lung3D, which includes the 3D models of 800 labeled pulmonary segments and their corresponding airways, arteries, veins, and intersegmental veins. We demonstrate that the proposed approach outperforms existing methods, providing a new perspective for pulmonary segment reconstruction. Code and data will be available at https://github.com/HINTLab/ImPulSe.

Renormalization group based implicit function approach to connecting orbits

Abstract This paper develops a comprehensive two-dimensional generalisation of the recently introduced Friction with Bristle Dynamics (FrBD) framework for rolling contact problems. The proposed formulation extends the one-dimensional FrBD model to accommodate simultaneous longitudinal and lateral slips, spin, and arbitrary transport kinematics over a finite contact region. The derivation combines a rheological representation of the bristle element with an analytical local sliding-friction law. By relying on an application of the Implicit Function Theorem, the notion of sliding velocity is then eliminated, and a fully dynamic friction model, driven solely by the rigid relative velocity, is obtained. Building upon this local model, three distributed formulations of increasing complexity are introduced, covering standard linear rolling contact, as well as linear and semilinear rolling in the presence of large spin slips. For the linear formulations, well-posedness, stability, and passivity properties are investigated under standard assumptions. In particular, the analysis reveals that the model preserves passivity under almost any parametrisation of practical interest. Numerical simulations illustrate steady-state action surfaces, transient relaxation phenomena, and the effect of time-varying normal loads. The results provide a unified and mathematically tractable friction model applicable to a broad class of rolling contact systems.

The paper discusses the basic and advanced properties of the Lambert W function in the complex domain. Despite many works devoted to this classical topic (dealing mainly with the principal branch of the function), several important questions have not been discussed or answered yet. Therefore, the aim of this paper is twofold. First, it describes the real and imaginary parts of all the complex branches of the Lambert W function via certain implicit equations (in the real domain). Second, applying such implicit characterizations strengthens and extends the existing knowledge on the key topics of the Lambert W function theory, such as the distribution and localization of its function values. Our findings in both research directions also offer a new insight into another important topic, namely, approximations of the Lambert W function values.

This manuscript investigates the time-fractional stochastic Keller-Segel-Navier-Stokes system in Hilbert space. This work provides a theoretical framework for analyzing cell migration by incorporating memory effects and environmental noise into the chemotactic signaling and fluid interaction. The proposed system captures key dynamics of cells respond to external gradients during directed movement. The existence of local and global mild solutions with uniqueness is studied under suitable conditions by using Banach fixed point and Banach implicit function theorem. The results are obtained in the pth moment by employing fractional calculus, stochastic analysis and Mittag-Leffler functions. Furthermore, we investigated the asymptotic stability of the proposed system as time approaches infinity.

This paper proposes a few methods of stratification for two study variables, contributing to the extensive body of research on optimum stratification in stratified sampling, for a known model-based allocation. The methods of finding points that stratify a heterogeneous population optimally have been obtained in the form of equations. These equations, which yield the optimal points for stratification (OPS), for the model-based allocation, are derived by minimizing the determinant of the variance-covariance matrix of stratified sampling for two study variables. However, these equations are lengthy as well as implicit, making them difficult and cumbersome for practical implementation. Therefore, the equations are algebraically and analytically transformed to derive a few methods for obtaining approximately optimal points for stratification (AOPS). The efficiencies of all the proposed methods are empirically evaluated using a few generated populations with results showing consistently efficient performance in stratifying populations optimally. The equations yielding OPS and their transformed methods yielding AOPS have demonstrated similar efficiencies in their performance. This suggests that the simpler and easy-to-use AOPS methods can serve as effective substitutes for the implicit and lengthy equations. All the methods proposed herein are derived under simple random sampling with replacement (SRSWR) design, but they are found true in simple random sampling without replacement (SRSWOR) provided finite population correction is neglected.

Accurate and transferable exchange-correlation (XC) functionals are central to the predictive power of density functional theory (DFT). However, conventional parameter optimization of XC functionals is typically performed using single-objective or stepwise strategies, which may lead to imbalanced performance across chemically diverse systems. This work introduces a multi-objective optimization framework, termed EBI4MO (explicit-by-implicit for multi-objectives), that enables simultaneous and consistent optimization with respect to multiple performance criteria. EBI4MO constructs a hierarchy of implicit functions that couple interdependent parameter groups across objectives, allowing sequential yet interlinked parameter updates. As a demonstration, EBI4MO is applied to optimize the parameters in hybrid XC functionals with dispersion corrections, using the GMTKN55 benchmark database. Two objectives are considered: minimizing the overall prediction error and achieving uniform improvement relative to B3LYP-D3(BJ), a widely used and balanced functional. The resulting functionals demonstrate consistent and balanced performance across all benchmark subsets, outperforming functionals optimized via conventional single-objective or stepwise methods. These results highlight the effectiveness and generality of EBI4MO, offering a new strategy for functional development and broader multi-objective optimization problems in computational chemistry.

A Physics-Informed Recurrent Neural Network for Long Sequence Remaining Useful Life Prediction of Rolling Bearing with Implicit Function

An implicit parallel adaptive momentum equation method for singularity issues in gas-liquid two-phase flows

In aerospace engineering, the wing is a critical factor in assessing the structural reliability of an aircraft. As the fundamental structural element of an aircraft, the reliability of the wing is of paramount importance to the aircraft’s flight performance and the safety of its occupants. Consequently, a reliability analysis of the wing represents a crucial avenue for enhancing the overall flight performance and safety of the aircraft. The paper examines the reliability of the double-beam wing structure of an aircraft in the event of failure under a range of potential scenarios. This study improves the response surface proxy modeling method by incorporating the principle of maximum entropy, thus optimizing its integration with the finite element analysis (FEA) method by reducing sampling needs and computational costs while enhancing accuracy for nonlinear implicit functions. Challenges included handling correlated multi-failure modes, addressed through series system analysis. This new approach allows for the reliability analysis of complex wing structures to be carried out in a more efficient manner.