Implicit Function Research Articles

Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations

This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. The fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom. We also assume that the fluid is subject to a uniform gravitational force that can be resolved into a sum of a vertical component and a component lying in the direction of the traveling wave velocity. This configuration arises, for instance, in the modeling of fluid flow down an inclined plane. We also study the effect of periodicity by allowing the fluid cross section to be periodic in various directions. The horizontal component of the gravitational field gives rise to stationary solutions that are pure shear flows, and we construct our solutions as perturbations of these by means of an implicit function argument. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces, including that these spaces are an algebra in the supercritical regime, which may be of independent interest.

Implicit function characterization of the curvilinear mask to realize parametric optical proximity correction with a neighborhood parallel tabu search.

The curvilinear mask structures provide significant benefits in improving lithographic resolution. Curvilinear masks, as opposed to rectilinear masks, have a wider range of structure types that can be used precisely to correct the contour of diffraction at sharp technological nodes. However, the curvilinear structure also makes the inverse design of mask in optical proximity correction (OPC) flow difficult. The current OPC of curvilinear masks uses pixel-based inverse optimization, which is extremely computationally intensive and takes up a lot of design data storage space. This paper proposes an implicit function to represent a large number of curve types with a small number of parameters to reduce computational complexity and the R&D cycle. Therefore, the ultra-high dimensional pixel-based OPC problem is transformed into a low-dimensional parameter search problem in the critical diffraction area of the mask pattern. The tabu search algorithm and neighborhood parallel computing strategy are then used to quickly search for optimal characterized parameters. The results of the simulation show that the parametric curvilinear OPC method achieves a higher image fidelity than that of rectilinear OPC. At the same time, it addresses the shortcomings of the traditional pixelated curvilinear mask OPC method, including high computational complexity, low manufacturability, and storage space occupancy.

Exploring the Connections Between the Implicit Function Theorem and Optimization Methods

Exploring the Connections Between the Implicit Function Theorem and Optimization Methods

Cryptanalysis of ARX-based White-box Implementations

At CRYPTO’22, Ranea, Vandersmissen, and Preneel proposed a new way to design white-box implementations of ARX-based ciphers using so-called implicit functions and quadratic-affine encodings. They suggest the Speck block-cipher as an example target.In this work, we describe practical attacks on the construction. For the implementation without one of the external encodings, we describe a simple algebraic key recovery attack. If both external encodings are used (the main scenario suggested by the authors), we propose optimization and inversion attacks, followed by our main result - a multiple-step round decomposition attack and a decomposition-based key recovery attack.Our attacks only use the white-box round functions as oracles and do not rely on their description. We implemented and verified experimentally attacks on white-box instances of Speck-32/64 and Speck-64/128. We conclude that a single ARX-round is too weak to be used as a white-box round.

Construction of controllability function as time of motion

This article is devoted to the controllability function method in admissible synthesis problems for linear canonical systems. The work considers methods of constructing such control so that the controllability function is time of motion of an arbitrary point to the origin. A canonical controlled system of linear equations $\dot{x}_i=x_{i+1}, i=\overline{1,n-1}, \dot{x}_n=u$ with control constraints $|u| \le d$ is considered. The controllability function $\Theta$ can be found as the only positive solution of the implicit equation $2a_0\Theta=(D(\Theta)FD(\Theta)x,x)$, where $D(\Theta)= diag(\Theta^ {-\frac{-2n-2i+1}{2}})_{i=1}^n$. Matrix $F=\{f_{ij}\}_{i,j=1}^n$ is positive definite and $a_0>0$ is chosen so that the control constraints are satisfied. The controllability function is motion time if $\dot{\Theta}= -1$. From this condition, an equation is obtained, the solution of which is considered in this work. Unlike previous works on this topic, no additional restrictions are imposed on the appearance of matrix $F$. The task of this article is to find the parameters set of the matrix $F$ and the column vector $a$, which satisfy the obtained equation and for which the controllability function is the time of movement from the point $x$ to the origin. In this way, we get a family of controls depending on this parameters such that the trajectory of system steers the origin in finite time. In general case, difficulties may arise when finding the solution of Cauchy problem of the corresponding system. Canonical system can be reduced to Euler's equation, for which a characteristic equation can be found, and therefore a trajectory in an explicit form. Two-dimensional, three-dimensional and four-dimensional canonical systems are considered. In each case, the matrix equation is solved and sets of parameters for which the controllability functions value will be the time of movement of an arbitrary point to the origin are found. Conditions on parameters are obtained from positive definiteness of the matrix $F$. Some parameters and an arbitrary initial point are chosen and the solution of Cauchy problem in analytical form is found.

Predicting continuum breakdown with deep neural networks

The multi-scale nature of gaseous flows poses tremendous difficulties for theoretical and numerical analysis. The Boltzmann equation, while possessing a wider applicability than hydrodynamic equations, requires significantly more computational resources due to the increased degrees of freedom in the model. The success of a hybrid fluid-kinetic flow solver for the study of multi-scale flows relies on accurate prediction of flow regimes. In this paper, we draw on binary classification in machine learning and propose the first neural network classifier to detect near-equilibrium and non-equilibrium flow regimes based on local flow conditions. Compared with classical semi-empirical criteria of continuum breakdown, the current method provides a data-driven alternative where the parameterized implicit function is trained by solutions of the Boltzmann equation. The ground-truth labels are derived rigorously from the deviation of particle distribution functions and the approximations based on the Chapman-Enskog ansatz. Therefore, no tunable parameter is needed in the criterion. Following the entropy closure of the Boltzmann moment system, a data generation strategy is developed to produce training and test sets. Numerical analysis shows its superiority over simulation-based samplings. A hybrid Boltzmann-Navier-Stokes flow solver is built correspondingly with an adaptive partition of local flow regimes. Numerical experiments including the one-dimensional Riemann problem, shear flow layer, and hypersonic flow around a circular cylinder are presented to validate the current scheme for simulating cross-scale and non-equilibrium flow physics. The quantitative comparison with a semi-empirical criterion and benchmark results demonstrates the capability of the current neural classifier to accurately predict continuum breakdown. The code for the data generator, hybrid solver, and neural network implementation is available in the open source repositories [1,2].

G2IFu: Graph-based implicit function for single-view 3D reconstruction

As the demand for 3D models increases, there is growing interest in reconstructing 3D objects from images using AI. In this paper, we propose G2IFu, a graph-based implicit function that successfully reconstructs a highly detailed 3D object mesh from a single image. Unlike previous methods that learn implicit functions with points, G2IFu aims to map graphs to implicit values. We make the following contributions: (1) Compared to independent 3D points, graphs have a larger perception space and contain specific spatial structure information. Therefore, we extend a 3D point p to a graph Gp by generating hypothesis points and establishing edges between them. We then predict the corresponding implicit value using a graph convolution network. Our experiments show that this method can effectively improve the prediction accuracy of implicit functions. (2) We introduce a prior boundary loss based on Gp to make the network pay more attention to the “key” points near the shape surface. To the best of our knowledge, G2IFu is the first model that introduces a graph into neural implicit representation. (3) Inspired by previous methods, we utilize the image’s global and local features to initialize Gp. We also introduce a self-attention module into G2IFu for better performance. We conduct experiments on the ShapeNet dataset and demonstrate that G2IFu can generate higher-quality 3D object shapes than previous single-view reconstruction methods. Additionally, we extend G2IFu to multi-view 3D reconstruction and achieve good performance.

Computing symmetries of implicit algebraic surfaces

We present a complete algorithm to compute the rotational, axial, reflectional and central symmetries of an algebraic surface defined by means of its implicit equation. The algorithms rely on the fact that the symmetries of the surface satisfy an algebraic condition verified by the polynomial defining the surface, and use a variety of ideas, some of them extensions to space of ideas already used to compute the symmetries of planar implicit curves, and some of them new. We flesh out our algorithms with several examples.

Optimization of Control Parameters of Magnetorheological Squeeze Film Dampers to Minimize the Vibration Amplitude of Rotors Passing the Critical Speed

Rotors are often coupled with a stationary part by rolling element bearings. To suppress their excessive vibration, the bearings are inserted in squeeze film dampers. The control of damping in the support elements offers the possibility to minimize the oscillation amplitude of accelerating or decelerating rotors, passing the regions of critical speeds. The controllable damping effect can be achieved if the squeeze film dampers are lubricated with magnetorheological oil. The change in the applied current feeding the electric coil changes magnetic induction in the damper gap, which changes the oil damping properties. The minimum vibration amplitude of the rotor running up or down through the resonance area is accomplished if the current increase or decrease is not sudden, but if it is distributed in some time interval. This article concentrates on determination of the optimum parameters of this manipulation. The developed procedure leads to solving an unconstrained optimization problem with the implicit objective function. The evolution method was used for its solving. In the investigated case, the proposed procedure made it possible to reduce maximum vibration amplitude by about 40% compared with the uncontrolled current decrease. The main contribution of the conducted research work is presentation of a new and original procedure for controlling the damping effect in the rotor supports. It provides a new idea to the designers and engineers regarding how to minimize amplitude of the rotor vibration when passing the critical speed. In addition, the article points to a new area of utilization of controllable magnetorheological squeeze film dampers.

Fast iterative method for local steric Poisson–Boltzmann theories in biomolecular solvation

This work proposes a fast iterative method for local steric Poisson–Boltzmann (PB) theories, in which the electrostatic potential is governed by the Poisson's equation and ionic concentrations satisfy equilibrium conditions. To present the method, we focus on a local steric PB theory derived from a lattice-gas model, as an example. The advantages of the proposed method in efficiency are achieved by treating ionic concentrations as scalar implicit functions of the electrostatic potential, though such functions are only numerically achievable. The existence, uniqueness, boundness, and smoothness of such functions are rigorously established. A Newton iteration method with truncation is proposed to solve a nonlinear system discretized from the generalized PB equations. The existence and uniqueness of the solution to the discretized nonlinear system are established by showing that it is a unique minimizer of a constructed convex energy. Thanks to the boundness of ionic concentrations, truncation bounds for the potential are obtained by using the extremum principle. The truncation step in iterations is shown to be energy and error decreasing. To further speed-up computations, we propose a novel precomputing-interpolation strategy, which is applicable to other local steric PB theories and makes the proposed methods for solving steric PB theories as efficient as for solving the classical PB theory. Analysis on the Newton iteration method with truncation shows local quadratic convergence for the proposed numerical methods. Applications to realistic biomolecular solvation systems reveal that counterions with steric hindrance stratify in an order prescribed by the parameter of ionic valence-to-volume ratio. Finally, we remark that the proposed iterative methods for local steric PB theories can be readily incorporated in well-known classical PB solvers.

Nilpotent algebras, implicit function theorem, and polynomial quasigroups

We prove an implicit function theorem for nilpotent (not necessarily associative or Lie) algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of the classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras.

Existence, Uniqueness and Ulam-Hyers-Rassias Stability of Differential Coupled Systems with Riesz-Caputo Fractional Derivative

Abstract This article deals with the existence, uniqueness and Ulam-Hyers--Rassias stability results for a class of coupled systems for implicit fractional differential equations with Riesz-Caputo fractional derivative and boundary conditions. We will employ the Banach’s contraction principle as well as Schauder’s fixed point theorem to demonstrate our existence results. We provide an example to illustrate the obtained results.

Velocity Obstacle for Polytopic Collision Avoidance for Distributed Multi-Robot Systems

Obstacle avoidance for multi-robot navigation with polytopic shapes is challenging. Existing works simplify the system dynamics or consider it as a convex or non-convex optimization problem with positive distance constraints between robots, which limits real-time performance and scalability. Additionally, generating collision-free behavior for polytopic-shaped robots is harder due to implicit and non-differentiable distance functions between polytopes. In this paper, we extend the concept of velocity obstacle (VO) principle for polytopic-shaped robots and propose a novel approach to construct the VO in the function of vertex coordinates and other robot's states. Compared with existing work about obstacle avoidance between polytopic-shaped robots, our approach is much more computationally efficient as the proposed approach for construction of VO between polytopes is optimization-free. Based on VO representation for polytopic shapes, we later propose a navigation approach for distributed multi-robot systems. We validate our proposed VO representation and navigation approach in multiple challenging scenarios including large-scale randomized tests, and our approach outperforms the state of art in many evaluation metrics, including completion rate, deadlock rate, and the average travel distance.

Implicit Function Theorems for Continuous Mappings and Their Applications

Implicit Function Theorems for Continuous Mappings and Their Applications

P-Regularity Theory and the Existence of a Solution to a Boundary Value Problem Continuously Dependent on Boundary Conditions

For a given boundary value problem, the existence of a solution depending continuously on the boundary conditions is analyzed. Previously, such a fact has been known only for the Cauchy problem, which is a classical result in the theory of differential equations. We prove a similar result for boundary value problems in the case when they are p-regular. In the general case, this result does not hold. Several implicit function theorems are proved in the degenerate case, which is a development of p-regularity theory concerning the existence of a solution to nonlinear differential equations. The results are illustrated by an example of a classical boundary value problem, namely, a degenerate Van der Pol equation is considered, for which the existence of a solution depending continuously on the boundary conditions of the perturbed problem is proved.

Random Solutions for Generalized Caputo Periodic and Non-Local Boundary Value Problems

In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving the generalized Caputo fractional derivative operator and supplemented with non-local and periodic boundary conditions. We make use of the fixed point theorems due to Banach and Krasnoselskii to derive the desired results. Examples illustrating the obtained results are also presented.

Implicit-implicit and parametric-implicit surface intersections in Euclidean n-space

In this paper, we study the intersection problem of two surfaces in which at least one surface is given by its implicit equation in Euclidean n-space. We first obtain the curvatures of the transversal intersection curve of two implicit surfaces. Later, the intersection problem is studied for a parametric surface and an implicit surface in n-dimensional Euclidean space. Finally, we present examples as applications of the given methods.

Implicit function‐based continuous representation for meticulous segmentation of cracks from high‐resolution images

AbstractHigh‐resolution (HR) crack images offer more detailed information for evaluating the structural condition and formulating effective maintenance or rehabilitation plans. However, the meticulous segmentation of HR crack images has been a challenge due to the limitations of mainstream deep learning algorithms that extract features in a discrete manner, as well as the constraints of computing resources. To address this issue, a novel implicit function‐integrated architecture, called the crack continuous refinement network (CCRN), was proposed for meticulous segmentation of cracks from HR images using a continuous representation manner. First, a crack continuous alignment module with a position encoding function was proposed to encode the tiny crack pixels that are easily lost in the sampling process. Then, a lightweight decoder embedded with implicit functions was customized to recover crack details from the aligned latent features and continuous position encoding information. Afterward, the gap between low‐resolution training images and HR inference results was bridged by the proposed continuous inference strategy. Finally, the robustness and practicability of the well‐trained CCRN were demonstrated by a parallel comparison and an unmanned aerial vehicle‐based field experiment.

Analysis and prediction of athlete's anxiety state based on artificial intelligence.

Obtaining athletes' anxiety accurately and regulating their psychological state helps improve their competitive performance. Therefore, this article uses a hierarchical clustering algorithm to identify the sources of stress of track and field athletes. A novel and efficient hierarchical clustering algorithm is proposed in this article. The algorithm consists of two stages: dividing and agglomerating. In the dividing stage, the initial data set is taken as a class and subclasses more than the actual number of clusters are obtained through multiple dividing. In the agglomerating phase, the subclasses divided during the dividing process are merged into the correct class. In addition, we construct an analysis model of athletes' anxiety state based on the radial basis function (RBF) model, where athletes' anxiety is divided into three categories: physical condition anxiety, competition state and cognitive state. The proposed model is trained from the official website of the China Track and Field Association. The athletes' information from 500 samples was arranged to form the sample database of athletes' data. The implicit unit center, function width and connection weight record the characteristics of various sports anxiety states. Then we used the Bayesian and Lagrange models as comparative models for evaluating the psychological state. Precision and efficiency were used for evaluation indexes. The proposed model's results are much better in accuracy and time than those of the Lagrange and Bayesian models. The outcome of the proposed research can provide a reasonable basis for the decision-making of stress relief for track and field athletes.

A transfer matrix method-based closed-form solution of sensitivities of dynamic properties and FRF for multi-span pipes under complex boundary conditions

The calculation of the first derivatives of dynamic properties and the frequency response functions with respect to design variables is a prerequisite for structural model updating and structural design. However, research about the sensitivity analysis of multi-span fluid-conveying pipes is still rarely performed. In this study, the closed-form formulae for the sensitivities are derived based on the transfer matrix method. Specifically, the difficulty in solving the sensitivities of the nonlinear eigenvalue problem is addressed from the characteristic determinant by invoking the implicit function theorem. One major advantage of the proposed method is that the closed-form formulas of sensitivities are given and no truncation errors are involved when compared with modal method and finite difference method. The method can explicitly deal with the analysis of the sensitivities of a pipe with arbitrary spans and accessories for complex boundary conditions. A three-span pipe is used to demonstrate the effectiveness and advantages of the method. The parametric eigenvalue sensitivity analysis provides useful design information and clamp looseness identification with structural model updating.