Smooth Maps Research Articles

A new definition of rough paths on manifolds

Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one loses the quantitative estimates controlling the convergence of the Picard sequence. Moreover, even with a definition of rough paths in smooth manifolds, ordinary and rough differential equations can only be solved locally in such case. In this paper, we first recall the foundations of the Lipschitz geometry, introduced in Rough Paths on Manifolds (Cass, T., Litterer, C. & Lyons, T.), along with the main findings that encompass the classical theory of rough paths in Banach spaces. Then we give what we believe to be a minimal framework for defining rough paths on a manifold that is both less rigid than the classical one and emphasized on the local behaviour of rough paths. We end by explaining how this same idea can be used to define any notion of coloured paths on a manifold.

Ramp-shaped neural tuning supports graded population-level representation of the object-to-scene continuum

We can easily perceive the spatial scale depicted in a picture, regardless of whether it is a small space (e.g., a close-up view of a chair) or a much larger space (e.g., an entire class room). How does the human visual system encode this continuous dimension? Here, we investigated the underlying neural coding of depicted spatial scale, by examining the voxel tuning and topographic organization of brain responses. We created naturalistic yet carefully-controlled stimuli by constructing virtual indoor environments, and rendered a series of snapshots to smoothly sample between a close-up view of the central object and far-scale view of the full environment (object-to-scene continuum). Human brain responses were measured to each position using functional magnetic resonance imaging. We did not find evidence for a smooth topographic mapping for the object-to-scene continuum on the cortex. Instead, we observed large swaths of cortex with opposing ramp-shaped profiles, with highest responses to one end of the object-to-scene continuum or the other, and a small region showing a weak tuning to intermediate scale views. However, when we considered the population code of the entire ventral occipito-temporal cortex, we found smooth and linear representation of the object-to-scene continuum. Our results together suggest that depicted spatial scale information is encoded parametrically in large-scale population codes across the entire ventral occipito-temporal cortex.

Environmental sustainability, nonlinear dynamics and chaos reloaded: 0 matters!

In this paper, we reconsider the overlapping generations (OLG) environmental model introduced in John and Pecchenino (1994) and Zhang (1999) by adopting the specification of the environmental dynamics proposed by Naimzada and Sodini (2010). The model is characterized by two different regimes that may alternate: one in which environmental evolution and economic growth co-evolve in the same direction; the other in which the environmental problem is not internalized by the agents, that is, the agents do not devolve any private resource to the environmental quality, leading to a possible trade-off between environment and economic growth. The analysis of the equilibrium dynamics, described by a two dimensional piecewise smooth map, shows that starting from a parametric configuration in which the dynamics are eventually driven by a unique regime, the increase of the negative effect of the agents’ consumption activity ends up in scenarios where the two regimes alternate, determining the rise of stable cycles or the occurrence of chaotic regimes. It is interesting to notice that because of the nonsmoothness of the map, the rise in environmental harm produced by economic activity may induce a sudden transition to chaotic regime.

Bifurcation Analysis of a Piecewise Smooth Map with Two Asymptotes

In this paper, we consider a discontinuous piecewise smooth system involving four parameters and two asymptotes, recently introduced as a model in engineering sciences. We classify and investigate its bifurcation behaviour. A local bifurcation analysis of the system in the range of parameters which has not been studied so far is undertaken and then supported by numerical computations. This reveals the existence of a flip bifurcation depends on the power singularity. Moreover, we state that a set of positive measure of points with divergent dynamic behaviour exists.

Geometrically Constrained Trajectory Optimization for Multicopters

In this article, we present an optimization-based framework for multicopter trajectory planning subject to geometrical configuration constraints and user-defined dynamic constraints. The basis of the framework is a novel trajectory representation built upon our novel optimality conditions for unconstrained control effort minimization. We design linear-complexity operations on this representation to conduct spatial–temporal deformation under various planning requirements. Smooth maps are utilized to exactly eliminate geometrical constraints in a lightweight fashion. A variety of state-input constraints are supported by the decoupling of dense constraint evaluation from sparse parameterization and the backward differentiation of flatness map. As a result, this framework transforms a generally constrained multicopter planning problem into an unconstrained optimization that can be solved reliably and efficiently. Our framework bridges the gaps among solution quality, planning efficiency, and constraint fidelity for a multicopter with limited resources and maneuvering capability. Its generality and robustness are both demonstrated by applications to different flight tasks. Extensive simulations and benchmarks are also conducted to show its capability of generating high-quality solutions while retaining the computation speed against other specialized methods by orders of magnitude.

Short-Time Existence for Harmonic Map Heat Flow with Time-Dependent Metrics

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by Li-Tam [The heat equation and harmonic maps of complete manifolds, Invent. Math., 1991] and Chen-Zhu [Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geometry, 2006]. In particular, we prove the short time existence of harmonic map heat flow along a complete Ricci flow $g(t)$ on $M$ into a complete manifold with curvature bounded from above with a smooth initial map of uniformly bounded energy density, under the assumptions that $|\text{Rm}(g(t))|\leq a/t$ and $g(t)$ is uniformly equivalent to $g(0)$.

Smooth dependence of fixed points of Hamiltonians

We consider the Lie semigroup of symplectic Hamiltonians acting on the open convex cone of positive definite matrices via linear fractional transformations. Each member of its interior contracts strictly the invariant Finsler metric, the Thompson metric on the cone, and has a unique positive definite fixed point. We show that the fixed point map is smooth. As applications, we obtain the smooth dependence of the solutions of discrete algebraic Riccati equations and a family of smooth maps from the Siegel upper half-plane over the cone of positive definite matrices into its imaginary part.

Multiscale aggregation network via smooth inverse map for crowd counting

Multiscale aggregation network via smooth inverse map for crowd counting

Einstein-Type Structures, Besse’s Conjecture, and a Uniqueness Result for a $$\varphi $$-CPE Metric in Its Conformal Class

In this paper, we study an extension of the CPE conjecture to manifolds M which support a structure relating curvature to the geometry of a smooth map \(\varphi : M \rightarrow N\). The resulting system, denoted by (\(\varphi \)-CPE), is natural from the variational viewpoint and describes stationary points for the integrated \(\varphi \)-scalar curvature functional restricted to metrics with unit volume and constant \(\varphi \)-scalar curvature. We prove both a rigidity statement for solutions to (\(\varphi \)-CPE) in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting (\(\varphi \)-CPE) with \(\varphi \) a harmonic map.

Near Tangent Dynamics in a Class of Hamiltonian Impact Systems

Tangencies correspond to singularities of impact systems, separating between impacting and non-impacting trajectory segments. The closure of their orbits constitutes the singularity set, which, even in the simpler billiard limit, is known to have a complex structure. The properties of this set are studied in a class of near integrable two degrees-of-freedom Hamiltonian impact systems. For this class of systems, in the integrable limit, on iso-energy surfaces, tangency appears at an isolated torus. We construct a piecewise smooth iso-energy return map for the perturbed flow near such a tangent torus. Away from the singularity set, this map has invariant curves, so, the singularity set is included in a limiting singularity band. An asymptotic upper bound of this band width is found for both non-resonant and resonant tangent tori. In the Diophantine case the band extent to both the impacting and nonimpacting regimes is, asymptotically, of order (\epsilon). In the resonance case its asymptotic extent to the nonimpacting regime is at most of order (\sqrt\epsilon) whereas its maximal asymptotic extent to the impacting regime is of order (\epsilon^2/3). Studying numerically a truncated standard form of the return map which is integrable at (\epsilon=0) and has a square root singularity with a singularity parameter (\alpha) reveals that only the strongest resonance obeys this maximal extent resonance scaling. We find that for all but one resonant case the singularity band has connecting orbits from its lower to its upper boundaries. The number of iterates required for an orbit to visit close to both the lower and upper boundaries of the singularity band diverges as (\epsilon\alpha^2) is decreased, giving rise to a multitude of transient complex phenomena. On the other hand, large (\alpha) values at a fixed small (\epsilon) lead to a significant reduction in this connection time.

An overview of generalised Kac-Moody algebras on compact real manifolds

A generalised notion of Kac-Moody algebra is defined using smooth maps from a compact real manifold M to a finite-dimensional Lie group, by means of complete orthonormal bases for a Hermitian inner product on the manifold and a Fourier expansion. The Peter–Weyl theorem for the case of manifolds related to compact Lie groups and coset spaces is discussed, and appropriate Hilbert bases for the space L2(M) of square-integrable functions are constructed. It is shown that such bases are characterised by the representation theory of the compact Lie group, from which a complete set of labelling operator is obtained. The existence of central extensions of generalised Kac-Moody algebras is analysed using a duality property of Hermitian operators on the manifold, and the corresponding root systems are constructed. Several applications of physically relevant compact groups and coset spaces are discussed.

Sub-Surface Geotechnical Data Visualization of Inaccessible Sites Using GIS

Geotechnical investigation, in hilly areas, for high-rise projects, becomes a problematic issue and costly process due to difficulties in mobilization and assembling the drilling equipment on mountainous terrains. The objective of this study is to map soil properties of study areas, especially at inaccessible sites, for reconnaissance. Digital soil maps for Tehsil Murree, Pakistan, have been developed using the emerging Geographical Information System (GIS). The research work involved the creation of an exhaustive database, by collecting and rectifying geotechnical data, followed by the digitization of the acquired data through integration with GIS, in an attempt to visualize, analyze and interpret the collected geotechnical information spatially. The soil data of 205 explanatory holes were collected from the available geotechnical investigation (GI) reports. The collection depth of soil samples, which were initially used for the design of deep and shallow foundations by different soil consultancies in the Murree area, was approximately 50 ft. below ground level. Appropriate spatial interpolation methods (i.e., the Kriging) were applied for the preparation of smooth surface maps of soil standard penetration tester number values, soil type and plasticity index. The accuracy of developed SPT N value and plasticity maps were then evaluated using the linear regression method, in which the predicted values of soil characteristics from developed maps and actual values were compared. SPT N value maps were developed up to a depth of 9.14 m below ground level and at every 1.52 m interval. The depth of refusal was considered in the developed maps. Soil type and plasticity maps were generated up to 15.24 m depth, again at every 1.52 m intervals, using color contours, considering the maximum predicted foundation depth for high-rise projects. The study has implications for academics and practitioners to map the soil properties for inaccessible sites using GIS, as the resulting maps have high accuracy.

Triharmonic CMC Hypersurfaces in Space Forms with at Most 3 Distinct Principal Curvatures

A k-harmonic map is a critical point of the k-energy defined on the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $$M^{n} (n\ge 3)$$ is a CMC proper triharmonic hypersurface with at most three distinct principal curvatures in a space form $$\mathbb {R}^{n+1}(c)$$ , then M has constant scalar curvature. This supports the generalized Chen’s conjecture when $$c\le 0$$ . When $$c=1$$ , we give an optimal upper bound of the mean curvature H for a non-totally umbilical proper CMC k-harmonic hypersurface with constant scalar curvature in a sphere. As an application, we give the complete classification of the 3-dimensional complete proper CMC triharmonic hypersurfaces in $$\mathbb {S}^{4}$$ .

Big or small? A new economic geography model with an endogenous switch in the market structure

We present a new economic geography (NEG) model with a linear demand function where firms may change the perception of their relative dimension with respect to the local market. If they perceive themselves as big, they behave as Cournot oligopolists, otherwise they behave as monopolistic competitors. We first compare the pure cases in which only one market form prevails in the two-region economy. Comparing these pure cases of monopolistic and oligopolistic competition only a quantitative difference emerges. Subsequently, we assume that firms switch behavior and start to interact strategically when the number of local firms is below a threshold; in that case, the market form evolves endogenously. Results change substantially. ‘Break’ and ‘sustain’ points are separated as in standard NEG models with an isoelastic demand function leading to a co-existence of equilibria. Stable partial agglomeration and oscillations with small amplitude are possible. The dynamics are described by a 1D piecewise smooth map, which can be continuous (in the pure monopolistic or oligopolistic cases) or discontinuous (when the market structure evolves endogenously). We analyze the bifurcation structure of the parameter space of the map comparing these cases. We show that the continuous maps have rather standard dynamics, while the discontinuous map is characterised by border collision bifurcations of fixed points and cycles, which lead to rich and complex bifurcation structures.

Lifting Generic Maps to Embeddings. Triangulation and Smoothing

We show that if a non-degenerate PL map f : N → M lifts to a topological embedding in then it lifts to a PL embedding in there. We also show that if a stable smooth map Nn → Mm, m ≥ n, lifts to a topological embedding in , then it lifts to a smooth embedding in there.

On ranges of set-valued mappings

We derive conditions ensuring that the range of a set-valued mapping with a compact convex domain covers a prescribed set. In Fréchet spaces, we consider approximations by one single-valued mapping such that the inverse of it has convex fibers. Subsequently, in Banach and finite-dimensional spaces, we focus on approximations determined by a convex set of bounded linear mappings such as Páles-Zeidan Jacobian, Clarke's generalized Jacobian, shields by T.H. Sweetser, or Neumaier's interval extensions of the derivative of a smooth mapping. As easy corollaries in Euclidean spaces, we obtain perturbation stability of the property of metric semiregularity under the additive perturbation by a single-valued mapping having sufficiently small calmness modulus; as well as the non-smooth Lyusternik-Graves theorem and Robinson's theorem by A.F. Izmailov. Finally, given two quadratic mappings f and g, a polyhedral convex set Ω, and an ordered interval Θ, we provide conditions guaranteeing that an ordered interval Γ is such that for each y∈Γ there is an x∈Ω with y=f(x) and g(x)∈Θ. This theorem has direct applications in power network security management such as preventing the electricity blackout.

Some classical analysis results for continuous definable mappings

In this paper, we show that some fundamental results for smooth mappings (e.g., the Brouwer degree formula, the implicit function and inverse function theorems, the mean value theorem, Sard's theorem, Hadamard's global invertibility criteria, Pourciau's surjectivity and openness results) have natural extensions for continuous mappings that are definable in o-minimal structures. The arguments rely on nice properties of definable mappings and sets.

Machine learning algorithms in geostatistical data analysis: formulation and observation

Geostatistical analyses on spatiotemporal datasets have been increasingly popular on diverse disciplines including environmental sciences, soil and earth studies, meteorology, hydrology, and oceanography. They help in documenting and summarizing information to understand the variation of the process and other parameters. Due to high spatial variability and noise in the dataset, traditional geostatistical tools often fail to produce desirable results. In view of this, the present study proposes two machine learning methods, namely artificial neural network (ANN) and k-nearest neighbour (kNN), along with the kriging interpolation to derive temporal distribution of soil parameters. Results in terms of smooth porosity maps demonstrate the effectiveness of the proposed algorithm.

Exposedness-Based Noise-Suppressing Low-Light Image Enhancement

A noise-suppressing low-light image enhancement approach is proposed in this paper based on the extent of exposedness at each image pixel. To this end, a progressive, structure-aware exposedness estimation procedure is presented that quantifies local and global exposedness. These exposedness values are leveraged to produce a locally smooth pixel-level map that signifies the required degrees of enhancement at image pixels. This map is subsequently used in an enhancement function, which satisfies a few important properties, to generate the enhanced image. Before the enhancement, inherent noise in the low-light image is diminished employing a detail-preserving, low gradient magnitude suppression method. Subjective and quantitative analysis of results on a wide variety of natural and synthetically generated low-light images from standard databases using PSNR, iRSE, SSIM, and measures of perceptual quality, natural image statistics and brightness preservation suggests that our approach in general outperforms the state-of-the-art. Ablation studies and further experiments show the importance of a few components of our approach, and that our approach is computationally fast.

Visible-NIR hyperspectral classification of grass based on multivariate smooth mapping and extreme active learning approach

Grass community classification is the basis for the development of animal husbandry and dynamic monitoring of environment, which has become a critical problem to further strengthen the intelligent management of grassland. Compared with grass survey based on satellite remote sensing, the visible near infrared (NIR) hyperspectral not only monitor dynamically in a short distance, but also have high dimensions and detailed spectral information in each pixel. However, the hyperspectral labeled sample for classification is expensive and manual selection is more subjective. In order to solve above limitations, we proposed a visible-NIR hyperspectral classification model for grass based on multivariate smooth mapping and extreme active learning (MSM–EAL). Firstly, MSM is used to preprocess and reconstruct the spectrum. Secondly, by jointing XGBoost and active learning (AL), the advanced samples with the largest amount of information are actively selected to improve the performance of target classification. Innovation lies in: (1) MSM global enhanced preprocessing spectral reconstruction algorithm is proposed, in which isometric feature mapping is effectively applied to the grass hyperspectral for the first time. (2) EAL framework is constructed to solve the issue of high cost and small number for hyperspectral labeled samples, at the same time, enhance the physical essence behind spectral classification more intuitively. A field hyperspectral collection platform is assembled to establish nm resolution visible-NIR hyperspectral dataset of grass, Grass1, containing 750 samples, which to verify the effectiveness of the model. Experiments on the Grass1 dataset confirmed that compared with the full spectrum, the time consumption of MSM was reduced by 9.471 s with guaranteed overall accuracy (OA). Comparing EAL with AL, and other classification algorithms, EAL improves OA 22.2% over AL, and XAL has the best performance value on Kappa, Macro, Recall and F1-score, respectively. Altogether, the lightweight MSM–EAL model realizes intelligent and real-time classification, providing a new method for obtaining high-precision inter group classification of grass.