Implicit Function Theorem Research Articles

Submersions, immersions, and étale maps in diffeology

Although structural maps such as subductions and inductions appear naturally in diffeology, one of the challenges is providing suitable analogues for submersions, immersions, and étale maps (i.e., local diffeomorphisms) consistent with the classical versions of these maps between manifolds. In this paper, we consider diffeological or plotwise versions of submersions, immersions, and étale maps as an adaptation of these maps to diffeology by a nonlinear approach. We study their diffeological properties from different aspects in a systematic fashion with respect to the germs of plots.In order to characterize the considered maps from their linear behaviors, we introduce a class of diffeological spaces, so-called diffeological étale manifolds, which not only contains the usual manifolds but also includes irrational tori. We state and prove versions of the rank and implicit function theorems, as well as the fundamental theorem on flows in this class. As an application, we use the results of this work to facilitate the computations of the internal tangent spaces and diffeological dimensions in a few interesting cases.

Neural network adaptive control for turbo‐generator of power systems with prescribed performance and unknown asymmetric actuator dead zone

AbstractThis paper investigates a neural network adaptive controller design method for turbo‐generator of power systems with external disturbance, unknown system dynamic, prescribed performance, and unknown actuator dead zone. First, the unknown system dynamic is estimated and overcome through a neural network. Using the implicit function theorem, the unknown asymmetric dead‐zone behavior of the actuator is overcome by another static neural network. Second, the external disturbance and the reconstruction error of neural networks are handled by a robust term updated online. Moreover, there is no requirement to pre‐know or off‐line estimate the reconstruction error of neural networks and the upper bound of the system external disturbance. Third, based on Lyapunov theory, the smooth control law is proposed. In the meantime, the uniform ultimate boundedness of the networks weights is strictly proved. Furthermore, the system state error satisfies the prescribed transient performance and can converge to a small neighborhood around zero. Finally, a numerical simulation shows the effectiveness of the proposed method.

Parametric Expansions of an Algebraic Variety Near Its Singularities II

The paper is a continuation and completion of the paper Bruno, A.D.; Azimov, A.A. Parametric Expansions of an Algebraic Variety Near Its Singularities. Axioms 2023, 5, 469, where we calculated parametric expansions of the three-dimensional algebraic manifold Ω, which appeared in theoretical physics, near its 3 singular points and near its one line of singular points. For that we used algorithms of Nonlinear Analysis: extraction of truncated polynomials, using the Newton polyhedron, their power transformations and Formal Generalized Implicit Function Theorem. Here we calculate parametric expansions of the manifold Ω near its one more singular point, near two curves of singular points and near infinity. Here we use 3 new things: (1) computation in algebraic extension of the field of rational numbers, (2) expansions near a curve of singular points and (3) calculation of branches near infinity.

Some qualitative analyses on a vegetation-water model with cross-diffusion and internal competition

This paper is concerned with a vegetation-water model with cross-diffusion and intra-plant competitive feedback under Neumann boundary conditions. First, we found that the equilibrium with small vegetation density is always unstable, and if the cross-diffusion coefficient is suitably large, the equilibrium with relatively large vegetation density loses its stability, and Turing instability occurs. A priori estimates of positive steady-state solutions are also established by the maximum principle of elliptic equations. Moreover, some qualitative analyses on the steady-state bifurcations for both simple and double eigenvalues are conducted in detail. Space decomposition and the implicit function theorem are used for double eigenvalues. In particular, the global continuation is obtained, and the result shows that there is at least one non-constant positive steady-state solution when cross-diffusion is large. Finally, numerical simulations are provided to prove and supplement theoretic research results, and some vegetation patterns with the increase of the soil water diffusion feedback intensity are formed, where the transition appears: gap [Formula: see text] stripe [Formula: see text] spot.

Gevrey asymptotic implicit function theorem

We prove an asymptotic implicit function theorem in the setting of Gevrey asymptotics with respect to a parameter. The unique implicitly defined solution admits a Gevrey asymptotic expansion and furthermore it is the Borel resummation of the corresponding implicitly defined formal power series solution. The main theorem can therefore be rephrased as an implicit function theorem for Borel summable power series. As an application, we give a diagonal or Jordan decomposition for holomorphic matrices in Gevrey asymptotic families.

Bifurcation and Spatiotemporal Patterns of SI Epidemic Model with Diffusion

This paper investigates a spatiotemporal SI epidemiological model under homogeneous Neumann boundary conditions. First, the long-time behavior of the solutions is described, a priori estimates of nonconstant positive solutions are given, and the nonexistence of nonconstant positive steady states is proved by the energy method. Second, the Turing instability of the positive constant steady-state is discussed, and the existence of nonconstant positive steady states is shown by using the degree theory. Moreover, applying the bifurcation theory, we establish the local and global structures of the steady-state bifurcation from simple eigenvalues, and describe some conditions for determining the direction of bifurcation, where the techniques of space decomposition and implicit function theorem are adopted to deal with the local structure of the steady-state bifurcation from double eigenvalues. Finally, some analysis results are supplemented by numerical simulations.

Equilibrium Figures for a Rotating Compressible Capillary Two-Layer Liquid

The paper proves the existence of a family of axisymmetric equilibrium figures as solutions of a stationary problem with unknown boundaries for the Navier–Stokes equations corresponding to the slow rotation of a viscous compressible two-layer liquid mass about some axis. It is assumed that the liquids are barotropic and capillary, and have different viscosities, the internal fluid being bounded by a closed surface. This interface does not intersect with the external boundary of the cloud. The proof is based on implicit function theorem and carried out in the Hölder spaces.

Bifurcation Solutions to the Templator Model in Chemical Self-Replication

In this paper, we are concerned with a diffusive templator model in chemical self-replication, which describes the process of an individual molecule duplicating itself. Firstly, the stability of non-negative constant equilibrium solution is introduced. Then the existence of Hopf bifurcation is proved. Particularly, the stability and the direction of Hopf bifurcation for the spatially homogeneous model are discussed. Furthermore, by space decomposition and implicit function theorem, it is shown that the system may undergo a steady-state bifurcation with a two-dimensional kernel. Finally, several numerical simulations are completed to demonstrate the theoretical results.

Extreme water level of surge chamber in hydropower plant under combined operating conditions

The design height of surge chambers is determined by the lowest and highest water levels under combined operating conditions. We considered four typical combined operating conditions: successive-load-rejection operating condition (SLROC), successive-load-acceptance operating condition (SLAOC), load-acceptance-then-rejection operating condition (LATROC), and load-rejection-then-acceptance operating condition (LRTAOC). A theoretical formula was derived to determine the worst superposition moment of extreme water levels using the implicit function theorem. An actual hydropower plant was used to analyse extreme water levels. The results showed that the tangential point for the headrace tunnel discharge-water level relationship curve of initial and superposed operating conditions was the worst superposition moment. The highest water level at the worst superposition moment was higher than that at the superposition moment of maximum flow into the surge chamber; similarly, the lowest water level at the worst superposition moment was lower than that at the superposition moment of maximum flow out of the surge chamber. When the cross-sectional area of impedance hole was small, the highest and lowest water levels occurred in SLROC and SLAOC, respectively. Conversely, when the cross-sectional area was large, the highest and lowest water levels occurred in LATROC and LRTAOC, respectively. The proposed formula avoids repeated trial calculations to determine extreme water levels, providing important theoretical significance and engineering application value.

Weak solutions for stochastic differential equations with additive fractional noise

We present a new approach to prove the existence of a weak solution for the equation dxt=f(t,xt)dt+g(t)dBtHwhere BtH is a fractional Brownian motion taking values in a separable Hilbert space, and f and g are suitable functions. Our idea involves using the implicit function theorem and the scaling property of the fractional Brownian motion to obtain a weak solution for the equation.

Analyzing inexact hypergradients for bilevel learning

Abstract Estimating hyperparameters has been a long-standing problem in machine learning. We consider the case where the task at hand is modeled as the solution to an optimization problem. Here the exact gradient with respect to the hyperparameters cannot be feasibly computed and approximate strategies are required. We introduce a unified framework for computing hypergradients that generalizes existing methods based on the implicit function theorem and automatic differentiation/backpropagation, showing that these two seemingly disparate approaches are actually tightly connected. Our framework is extremely flexible, allowing its subproblems to be solved with any suitable method, to any degree of accuracy. We derive a priori and computable a posteriori error bounds for all our methods and numerically show that our a posteriori bounds are usually more accurate. Our numerical results also show that, surprisingly, for efficient bilevel optimization, the choice of hypergradient algorithm is at least as important as the choice of lower-level solver.

NDARTS: A Differentiable Architecture Search Based on the Neumann Series

Neural architecture search (NAS) has shown great potential in discovering powerful and flexible network models, becoming an important branch of automatic machine learning (AutoML). Although search methods based on reinforcement learning and evolutionary algorithms can find high-performance architectures, these search methods typically require hundreds of GPU days. Unlike searching in a discrete search space based on reinforcement learning and evolutionary algorithms, the differentiable neural architecture search (DARTS) continuously relaxes the search space, allowing for optimization using gradient-based methods. Based on DARTS, we propose NDARTS in this article. The new algorithm uses the Implicit Function Theorem and the Neumann series to approximate the hyper-gradient, which obtains better results than DARTS. In the simulation experiment, an ablation experiment was carried out to study the influence of the different parameters on the NDARTS algorithm and to determine the optimal weight, then the best performance of the NDARTS algorithm was searched for in the DARTS search space and the NAS-BENCH-201 search space. Compared with other NAS algorithms, the results showed that NDARTS achieved excellent results on the CIFAR-10, CIFAR-100, and ImageNet datasets, and was an effective neural architecture search algorithm.

Why are the Solutions to Overdetermined Problems Usually “As Symmetric as Possible”?

In this paper, we study the symmetry properties of nondegenerate critical points of shape functionals using the implicit function theorem. We show that, if a shape functional is invariant with respect to some one-parameter group of rotations, then its nondegenerate critical points (bounded open sets with smooth enough boundary) share the same symmetries. We also consider the case where the shape functional exhibits translational invariance in addition to just rotational invariance. Finally, we study the applications of this result to the theory of one/two-phase overdetermined problems of Serrin-type. En passant, we give a simple proof of the fact that, under suitable smoothness assumptions, the ball is the only nondegenerate critical point of the Lagrangian associated to the maximization problem for the torsional rigidity under a volume constraint. We remark that the proof does not rely on either the method of moving planes or rearrangement techniques.

Nonclassical Parametric Variational Technique to Manipulability Control of a Serial-Link Robot That Is Used in Treatment of Femoral Shaft Fractures

Robot-assisted intramedullary nailing is a minimally invasive surgical procedure commonly used to treat femur fractures. Despite its benefits, there are several disadvantages associated with this technique, such as frequent malalignment, physical fatigue, and excessive radiation exposure for medical personnel. Therefore, it is crucial to ensure that robotic surgery for fracture reduction is precise and safe. Precise calculation and regulation of the robot’s reduction force are of utmost importance. In this study, we propose a manipulator that utilises robot assistance and indirect contact with the femur to effectively reduce fractures in the shaft. The dynamics of the reduction robot are analysed using the implicit function theorem, which allows us to address the reduced problem. A parametric approach is presented to tackle the initial algebraic constraints, enabling the approximation of the state-space solution while simultaneously controlling the class of constraints in a multiway manner. This approach simplifies the problem from an infinite-dimensional one to a finite-dimensional one, leading to an approximate solution obtained by solving a set of control linear algebraic equations. The proposed robotic-assisted system enhances fracture repositioning while reducing radiation exposure for both the patient and the medical staff. Through numerical results and their practical application, we have developed an efficient method that yields positive outcomes.

A new HEAP-inspired method for embedded power flows and path-related holomorphic properties

Power flow problems can be robustly solved by holomorphic embedding (HE) methods which require the holomorphic property of solution functions near a reference state and also along a path bridging this state and the target state, for determining the effectiveness of series expansion and physical realizability (PRL), respectively, where PRL means the complex conjugacy of solution values at the target state. Existing studies generally assume holomorphic properties beforehand or assert them from a convergence of series expansion or implicit function theorem at separate points, however, which can hardly be applied to confirm the holomorphic property along a path and then PRL (because a path has infinitely many points and cannot be numerically inspected point-by-point). To this end, a new approach PHVACR is enlightened by HEAP and employs an expansion in the rectangular form and also Cauchy–Riemann (CR) conditions to numerically verify holomorphic properties around a path in an arc-by-arc manner. Here, ‘HE’ is short for holomorphic embedding, ‘AP’ in HEAP indicates an arc-length parametrization, and ‘PHVA’ in PHVACR refers to the path-related holomorphy validation with ‘CR’ representing the Cauchy–Riemann conditions. To explain the importance of PHVACR and its performance, counter-intuitive examples are discussed to explain the limitations of existing approaches, and large problems with 10,000+ buses are also simulated to examine the effectiveness and scalability of PHVACR which are compared with other HE methods.

Estimates of the size of the domain of the implicit function theorem: a mapping degree-based approach

Estimates of the size of the domain of the implicit function theorem: a mapping degree-based approach

Steady-State Bifurcation and Spatial Patterns of a Chemical Reaction System

This paper studies the Lengyel–Epstein chemical reaction system with nonlinear functional response and no-flux boundary conditions. We first investigate the existence of steady-state bifurcation solutions of the system. Then, the stability of bifurcation solutions is analyzed. Meanwhile, some spatial patterns induced by steady-state bifurcation are simulated numerically and depicted graphically. It is well known that the classical bifurcation theory in nonlinear dynamical systems is based on simple eigenvalues. However, this is not always the case. Sometimes, the kernel of an objective operator is of two or more dimensions. For such cases, there is no existing theory to deal with them. In this paper, by using the space decomposition technique and implicit function theorem, we analyze the bifurcation phenomenon of the system in the case of a two-dimensional kernel of some certain objective operator. The results show that the chemical reaction between activator iodide and inhibitor chlorite can proceed stably under certain conditions.

Robust [formula omitted] dynamic output feedback control for nonlinear time-varying delay rectangular descriptor Markov jump systems

The robust H∞ dynamic output feedback (DOF) control for a class of nonlinear time-varying delay rectangular descriptor Markov jump systems (DMJSs) is discussed in this paper, there are no restrictions on the rows and columns of the differential matrix E. The rectangular DOF controller is designed to make the closed-loop system as square DMJSs. Firstly, by utilizing a mode-dependent and delay-dependent Lyapunov-Krasovskii (L-K) functional and implicit function theorem, sufficient conditions are given to guarantee that the augmented closed-loop systems are robust stochastically admissible with an H∞ performance and have unique solution simultaneously. Then based on certain matrix decoupling techniques, a new strict linear matrix inequality (LMI) sufficient condition for the existence of a rectangular DOF controller is obtained, and the controller is given. Finally, numerical examples are presented to verify the validity of the results.

Qualitative properties, bifurcations and chaos of a discrete predator–prey system with weak Allee effect on the predator

In this work, the qualitative properties of the fixed points in the non-hyperbolic and degenerate cases, codimension-one bifurcations and Marotto’s chaos of a discrete predator–prey system with weak Allee effect on the predator are investigated. Utilizing the center manifold theorem and the reduction principle, the qualitative property of each fixed point in the non-hyperbolic case is explored. Based on the approximate flow theory, the qualitative property of the boundary fixed point in the degenerate case is investigated. Making use of the center manifold theorem and the bifurcation theory, all the potential codimension-one bifurcation types of the co-existence fixed point are explored, including flip bifurcation and Neimark–Sacker bifurcation. For each type of these bifurcations, not only the concise non-degenerate condition expressed by the system parameters is given, but also the analytical expression of the system orbit caused by the bifurcation is derived. Based on the non-negativity of the bifurcation critical values, the non-resonant and non-degenerate conditions for the occurrence of these bifurcations, the system parameter plane is divided into several regions using tools such as the polynomial complete discriminant system theory, real root separation theory, dichotomy method, implicit function theorem and Cardan’s formula. In each region, the direction and stability of these bifurcations are studied, which helps to theoretically explain the impact of Allee effect on the system. By proving that the co-existence fixed point is a snap-back repeller under appropriate conditions, it is obtained that the system is chaotic. Numerical simulations are completely consistent with all the theoretical analyses. The research results show that the impact of Allee effect on the population system depends not only on the assumption and construction of the system, but also on the system parameter and initial value of the system.

The Implicit Function Theorem in the Neighborhood of an Abnormal Point

The Implicit Function Theorem in the Neighborhood of an Abnormal Point