Smooth Maps Research Articles

Liouville type theorems and stability of [formula omitted]-harmonic maps

In this paper, we first raise the following question: can we obtain the p-stress energy tensor Sp that is associated with the p-energy functional Ep vanishes under some interesting conditions? This motivates us to introduce the notions of the ΦS,p-energy density eΦS,p(u), and the ΦS,p-energy functional EΦS,p(u) of a map u:M→N, that are related to the p-stress energy tensor Sp of a smooth map u between two Riemannian manifolds M and N. We derive the first variation formula of type I and type II, and the second variation formula for the ΦS,p-energy functional EΦS,p(u). We also introduce the stress energy tensor SΦS,p for the ΦS,p-energy functional EΦS,p, the notions of ΦS,p-harmonic maps, and stable ΦS,p-harmonic maps between Riemannian manifolds. Then we obtain some properties for weakly conformal ΦS,p-harmonic maps and horizontally conformal ΦS,p-harmonic maps, and prove some Liouville type results for ΦS,p-harmonic maps from some complete Riemannian manifolds under various conditions on the Hessian of the distance function and the asymptotic behavior of the map at infinity. By an extrinsic average variational method in the calculus of variations (Wei; 1989, 1983), we find ΦS,p-SSU manifolds and prove that any stable ΦS,p-harmonic map from or into a compact ΦS,p-SSU manifold (to or from a compact manifold) must be constant (cf. Theorems 5.1 and 6.1). We further prove that the homotopic class of any map from a compact manifold into a compact ΦS,p-SSU manifold contains elements of arbitrarily small ΦS,p-energy, and the homotopic class of any map from a compact ΦS,p-SSU manifold into a manifold contains elements of arbitrarily small ΦS,p-energy (cf. Theorems 7.1 and 8.2). As immediate consequences, we give a simple and direct proof of the above Theorems 5.1 and 6.1. These Theorems 5.1, 6.1, 7.1 and 8.2 give rise to the concept of ΦS,p-strongly unstable (ΦS,p-SU) manifolds, extending the notions of strongly unstable (SU), p-strongly unstable (p-SU), Φ-strongly unstable (Φ-SU) and ΦS-strongly unstable (ΦS-SU) manifolds (cf. Howard and Wei, 1986; Wei and Yau, 1994; Wei, 1998; Han and Wei, 2019; Feng et al., 2021). Hence, superstrongly unstable (SSU), p-superstrongly unstable (p-SSU), Φ-superstrongly unstable (Φ-SSU) and ΦS superstrongly unstable (ΦS-SSU) manifolds are strongly unstable (SU), p-strongly unstable (p-SU), Φ-strongly unstable (Φ-SU) and ΦS-strongly unstable (ΦS-SU) manifolds respectively, and enjoy their wonderful properties. We also introduce the concepts of ΦS,p-unstable (ΦS,p-U) manifold and establish a link of ΦS,p-SSU manifold to p-SSU manifold and topology. Compact ΦS,p-SSU homogeneous spaces are studied.

Simple approach to characterizing band topology in bosonic pairing Hamiltonians

We revisit the problem of characterizing band topology in dynamically-stable quadratic bosonic Hamiltonians that do not conserve particle number. We show this problem can be rigorously addressed by a smooth and local adiabatic mapping procedure to a particle number conserving Hamiltonian. In contrast to a generic fermionic pairing Hamiltonian, such a mapping can always be constructed for bosons. Our approach shows that particle non-conserving bosonic Hamiltonians can be classified using known approaches for fermionic models. It also provides a simple means for identifying and calculating appropriate topological invariants. We also explicitly study dynamically stable but non-positive definite Hamiltonians (as arise frequently in driven photonic systems). We show that in this case, each band gap is characterized by two distinct invariants.

Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

AbstractGiven any smooth Anosov map, we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that, in the case of expanding maps, it reduces exactly to the usual space of functions of bounded variation which has proved to be particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations, which provides new information that could prove useful in treating hyperbolic systems with singularities.

Modeling the relationship between malaria prevalence and insecticide-treated bed net coverage in Nigeria using a Bayesian spatial generalized linear mixed model with a Leroux prior

OBJECTIVESTo evaluate malaria transmission in relation to insecticide-treated net (ITN) coverage in Nigeria.METHODSWe used an exploratory analysis approach to evaluate variation in malaria transmission in relation to ITN distribution in 1,325 Demographic and Health Survey clusters in Nigeria. A Bayesian spatial generalized linear mixed model with a Leroux conditional autoregressive prior for the random effects was used to model the spatial and contextual variation in malaria prevalence and ITN distribution after adjusting for environmental variables.RESULTSSpatial smoothed maps showed the nationwide distribution of malaria and ITN. The distribution of ITN varied significantly across the 6 geopolitical zones (p<0.05). The North-East had the least ITN distribution (0.196±0.071), while ITN distribution was highest in the South-South (0.309±0.075). ITN coverage was also higher in rural areas (0.281±0.074) than in urban areas (0.240±0.096, p<0.05). The Bayesian hierarchical regression results showed a non-significant negative relationship between malaria prevalence and ITN coverage, but a significant spatial structured random effect and unstructured random effect. The correlates of malaria transmission included rainfall, maximum temperature, and proximity to water.CONCLUSIONSReduction in malaria transmission was not significantly related to ITN coverage, although much could be achieved in attempts to curtail malaria transmission through enhanced ITN coverage. A multifaceted and integrated approach to malaria control is strongly advocated.

Color Nameability Predicts Inference Accuracy in Spatial Visualizations

AbstractColor encoding is foundational to visualizing quantitative data. Guidelines for colormap design have traditionally emphasized perceptual principles, such as order and uniformity. However, colors also evoke cognitive and linguistic associations whose role in data interpretation remains underexplored. We study how two linguistic factors, name salience and name variation, affect people's ability to draw inferences from spatial visualizations. In two experiments, we found that participants are better at interpreting visualizations when viewing colors with more salient names (e.g., prototypical ‘blue’, ‘yellow’, and ‘red’ over ‘teal’, ‘beige’, and ‘maroon’). The effect was robust across four visualization types, but was more pronounced in continuous (e.g., smooth geographical maps) than in similar discrete representations (e.g., choropleths). Participants' accuracy also improved as the number of nameable colors increased, although the latter had a less robust effect. Our findings suggest that color nameability is an important design consideration for quantitative colormaps, and may even outweigh traditional perceptual metrics. In particular, we found that the linguistic associations of color are a better predictor of performance than the perceptual properties of those colors. We discuss the implications and outline research opportunities. The data and materials for this study are available athttps://osf.io/asb7n

Historical dynamics of the Chinese dynasties

We develop a framework for studying state division and unification, and as a case study we focus on modelling the territorial patterns in imperial China during periods of unity and upheaval. As a modelling tool we employ discrete dynamical systems and analyse two models: the logistic map and a new class of maps, which we name ren maps. The critical transitions exhibited by the models can be used to capture the process of territorial division but also unification. We outline certain limitations of uni-modal, smooth maps for our modelling purposes and propose ren maps as an alternative, which we use to reproduce the territorial dynamics over time. As a result of the modelling we arrive at a quantitative measure for asabiyyah, a notion of group solidarity, whose secular cycles match the historical record over 1800 years, from the time of the Warring States to the beginning of the Ming dynasty. Furthermore, we also derive an equation for aggregate asabiyyah which can be employed in other cases of interest.

A new cryptographic algorithm via a two-dimensional chaotic map

In this paper, a new cryptographic algorithm using a two-dimensional piecewise smooth nonlinear chaotic map is suggested. It depends on the confusion-diffusion model (permutation-substitution model). Firstly, the plain image is shuffled using the logistic map (confusion). Secondly, chaotic sequences are produced by two-dimensional piecewise smooth nonlinear chaotic map. Then, the produced sequences are preprocessed to integers between 0 and 255. Finally, the shuffled image is masked using the chaotic sequences (diffusion). The suggested algorithm is tested by different types of images. Furthermore, the decryption algorithm is implemented to retrieve the plain (original) image using the secret key. Experimental results and security analyses are applied using the suggested algorithm and its performances are validated with recent cryptographic algorithms. The security and performance analyses conclude that the suggested algorithm is secure, fast, and resists various attacks.

Large Deviations and Entropy Production in Viscous Fluid Flows

We study the motion of a particle in a random time-dependent vector field defined by the 2D Navier–Stokes system with a noise. Under suitable non-degeneracy hypotheses we prove that the empirical measures of the trajectories of the pair (velocity field, particle) satisfy the LDP with a good rate function. Moreover, we show that the law of a unique stationary solution restricted to the particle component possesses a positive smooth density with respect to the Lebesgue measure in any finite time. This allows one to define a natural concept of the entropy production, and to show that its time average is a bounded function of the trajectory. The proofs are based on a new criterion for the validity of the level-3 LDP for Markov processes and an application of a general result on the image of probability measures under smooth maps to the laws associated with the motion of the particle.

VLSI Design for High‐Precision Three‐Dimensional Depth Perception Chip

This paper presents a Very large scale integration (VLSI) design method for Three-dimensional (3D) depth perception chip based on infrared coding structure light. The primary sub-modules on the chip contain the speckle pattern preprocessing module, block-matching disparity estimation, depth mapping and post-processing. The chip employs pipelining technology, and after Application specific integrated circuit (ASIC) verification, it proves that our chip has more advantages in performance of depth precision (12bits, 1mm @ 1m), image resolution (1280×960), time delay (less than 17ms), range limit (0.4~6m). It also can generate more stable and smooth depth map in real-time, which can be used in 3D recognition, motion capture or scene perception.

Hidden dynamics for piecewise smooth maps

We develop a hidden dynamics formulation of regularisation for piecewise smooth maps. This involves blowing up the discontinuity into an interval, but in contrast to piecewise smooth flows every preimage of the discontinuity needs to be blown up as well. This results in a construction similar to classic approaches to the Denjoy counterexample.

The contact mappings of a flat (2,3,5)-distribution

Let Ω and Ω′ be open subsets of a flat (2,3,5)-distribution. We show that a C1-smooth contact mapping f: Ω → Ω′ is a $$C^\infty$$ -smooth contact mapping. Ultimately, this is a consequence of the rigidity of the associated stratified Lie group. (The Tanaka prolongation of the Lie algebra is of finite type.) The conclusion is reached through a careful study of some differential identities satisfied by components of the Pansu derivative of a C1-smooth contact mapping.

Graphical Convergence of Subgradients in Nonconvex Optimization and Learning

We investigate the stochastic optimization problem of minimizing population risk, where the loss defining the risk is assumed to be weakly convex. Compositions of Lipschitz convex functions with smooth maps are the primary examples of such losses. We analyze the estimation quality of such nonsmooth and nonconvex problems by their sample average approximations. Our main results establish dimension-dependent rates on subgradient estimation in full generality and dimension-independent rates when the loss is a generalized linear model. As an application of the developed techniques, we analyze the nonsmooth landscape of a robust nonlinear regression problem.

Projected and Near-Projected Embeddings

A stable smooth map f : N → M is called k-realizable if its composition with the inclusion M ⊂ M × ℝk is C0-approximable by smooth embeddings; and a k-prem if the same composition is C∞-approximable by smooth embeddings, or, equivalently, if f lifts vertically to a smooth embedding N ↪ M × ℝk. It is obvious that if f is a k-prem, then it is k-realizable. We refute the so-called “prem conjecture” that the converse holds. Namely, for each n = 4k+3 ≥ 15 there exists a stable smooth immersion Sn ↬ ℝ2n−7 that is 3-realizable but is not a 3-prem. We also prove the converse in a wide range of cases. A k-realizable stable smooth fold map Nn → M2n−q is a k-prem if q ≤ n and q ≤ 2k−3; or if q < n/2 and k = 1; or if q ∈ {2k−1, 2k−2} and k ∈ {2, 4, 8} and n is sufficiently large.

Smooth flat maps over commutative DG-rings

We study smooth maps that arise in derived algebraic geometry. Given a map \(A \rightarrow B\) between non-positive commutative noetherian DG-rings which is of flat dimension 0, we show that it is smooth in the sense of Toën–Vezzosi if and only if it is homologically smooth in the sense of Kontsevich. We then show that B, being a perfect DG-module over \(B\otimes ^{{\mathrm {L}}}_A B\) has, locally, an explicit semi-free resolution as a Koszul complex. As an application we show that a strong form of Van den Bergh duality between (derived) Hochschild homology and cohomology holds in this setting.

Motion Mappings for Continuous Bilateral Teleoperation

Mapping operator motions to a robot is a key problem in teleoperation. Due to differences between local and remote workspaces, such as object locations, it is particularly challenging to derive smooth motion mappings that fulfill different goals (e.g., picking objects with different poses on the two sides or passing through key points). Indeed, most state-of-the-art methods rely on mode switches, leading to a discontinuous, low-transparency experience. In this letter, we propose a unified formulation for position, orientation and velocity mappings based on the poses of objects of interest in the operator and robot workspaces. We apply it in the context of bilateral teleoperation. Two possible implementations to achieve the proposed mappings are studied: an iterative approach based on locally-weighted translations and rotations, and a neural network approach. Evaluations are conducted both in simulation and using two torque-controlled Franka Emika Panda robots. Our results show that, despite longer training time, the neural network approach provides faster mapping evaluations and lower interaction forces for the operator, which are crucial for continuous, real-time teleoperation.

Noise-modulated neural networks for selectively functionalizing sub-networks by exploiting stochastic resonance

In the phenomenon of stochastic resonance, adding a certain level of nonzero noise to a nonlinear system reduces information loss. A previous study proposed a neural network consisting of thresholding functions that exploit stochastic resonance at run time and during training, with the aim of smooth mapping and backpropagation. Such a neural network can be rephrased as one that operates only when noise is added, i.e., one that is unable to smoothly map and train when noise is absent. Focusing on both explanations simultaneously, a neural network for which only a sub-network is activated selectively by adding noise locally on that sub-network is proposed in this paper. To this end, a new activation function is introduced. It exploits stochastic resonance and presents null output and derivative when no noise is added. Simple simulations confirm that the proposed neural network with the new activation function allows the sub-network to be functionalized selectively, and interpolations are investigated by imposing varying noise intensity on various regions of the network after sub-networks are trained separately.

Transformations of Closed Invariant Curves and Closed-Invariant-Curve-Like Chaotic Attractors in Piecewise Smooth Systems

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve collides with a border but also via a homoclinic bifurcation. Furthermore, we describe an unusual transformation from a closed invariant curve to a large amplitude chaotic attractor and demonstrate that this transition occurs in two steps, involving a small amplitude closed-invariant-curve-like chaotic attractor.

Bach and Einstein’s equations in presence of a field

The aim of this paper is to introduce and justify a possible generalization of the classic Bach field equations on a four-dimensional smooth manifold [Formula: see text] in the presence of field [Formula: see text], given by a smooth map with source [Formula: see text] and target another Riemannian manifold. Those equations are characterized by the vanishing of a two times covariant, symmetric, traceless and conformally invariant tensor field, called [Formula: see text]-Bach tensor, that in absence of the field [Formula: see text] reduces to the classic Bach tensor, and by the vanishing another tensor related to the bi-energy of [Formula: see text]. Since solutions of the Einstein-massless scalar field equations, or more generally, of the Einstein field equations with source the wave map [Formula: see text] solves those generalized Bach’s equations, we include the latter in our analysis providing a systematic study for them, relying on the recent concept of [Formula: see text]-curvatures. We take the opportunity to discuss the related topic of warped product solutions.

General Convergence Analysis of Stochastic First-Order Methods for Composite Optimization

In this paper, we consider stochastic composite convex optimization problems with the objective function satisfying a stochastic bounded gradient condition, with or without a quadratic functional growth property. These models include the most well-known classes of objective functions analyzed in the literature: nonsmooth Lipschitz functions and composition of a (potentially) nonsmooth function and a smooth function, with or without strong convexity. Based on the flexibility offered by our optimization model, we consider several variants of stochastic first-order methods, such as the stochastic proximal gradient and the stochastic proximal point algorithms. Usually, the convergence theory for these methods has been derived for simple stochastic optimization models satisfying restrictive assumptions, and the rates are in general sublinear and hold only for specific decreasing stepsizes. Hence, we analyze the convergence rates of stochastic first-order methods with constant or variable stepsize under general assumptions covering a large class of objective functions. For constant stepsize, we show that these methods can achieve linear convergence rate up to a constant proportional to the stepsize and under some strong stochastic bounded gradient condition even pure linear convergence. Moreover, when a variable stepsize is chosen we derive sublinear convergence rates for these stochastic first-order methods. Finally, the stochastic gradient mapping and the Moreau smoothing mapping introduced in the present paper lead to simple and intuitive proofs.

The inverse-deformation approach to fracture

We propose a one-dimensional, nonconvex elastic constitutive model with higher gradients that can predict spontaneous fracture at a critical load via a bifurcation analysis. It overcomes the problem of discontinuous deformations without additional field variables, such as damage or phase-field variables, and without a priori specified surface energy. Our main tool is the inverse deformation, which can be extended to be a piecewise smooth mapping even when the original deformation has discontinuities describing cracks opening. We exploit this via the inverse-deformation formulation of finite elasticity due to Shield and Carlson, including higher gradients in the energy. The problem is amenable to a rigorous global bifurcation analysis in the presence of a unilateral constraint. Fracture under hard loading occurs on a bifurcating solution branch at a critical applied stretch level, with a sudden drop to zero stress. Fractured solutions are found to have sharply defined cracks with surface energy arising from higher gradient effects.