Representations Of Manifolds Research Articles

Topological Deformations of Manifolds by Algebraic Compositions in Polynomial Rings

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism.

Existence of the carrying simplex for a retrotone map

We present a dynamical approach to the study of unordered, attracting manifolds of retrotone maps commonly known as carrying simplices. Our approach is novel in that it uses the radial representation of unordered manifolds over the probability simplex coupled with distances between these manifolds measured by way of the Harnack and Hausdorff metrics. We establish Kuratowski convergence of radial representations of unordered manifolds to a unique function which then provides the locally Lipschitz radial representation of the carrying simplex.

Moment maps and isoparametric hypersurfaces in spheres — Grassmannian cases

We expect that every Cartan–Münzner polynomial of degree four can be described as a squared-norm of a moment map for a Hamiltonian action. Our expectation is known to be true for Hermitian cases, that is, those obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. In this paper, we prove that our expectation is true for the Cartan–Münzner polynomials obtained from the isotropy representations of Grassmannian manifolds of rank two over R, C or H. The quaternion cases are the first non-Hermitian examples that our expectation is verified.

Aerodynamic Optimization Framework for a Three-Dimensional Nacelle Based on Deep Manifold Learning-Assisted Geometric Multiple Dimensionality Reduction

As a core component of an aero-engine, the aerodynamic performance of the nacelle is essential for the overall performance of an aircraft. However, the direct design of a three-dimensional (3D) nacelle is limited by the complex design space consisting of different cross-section profiles and irregular circumferential curves. The deep manifold learning-assisted geometric multiple dimensionality reduction method combines autoencoders (AE) with strong capabilities for non-linear data dimensionality reduction and class function/shape function transformation (CST). A novel geometric dimensionality reduction method is developed to address the typical constraints of nacelle parameterization. Low-dimensional latent variables are extracted from the high-dimensional design space to achieve a parametric representation of 3D nacelle manifolds. Compared with traditional parametric methods, the proposed geometric dimensionality reduction method improves the accuracy and efficiency of geometric reconstruction and aerodynamic evaluation. A multi-objective optimization framework is proposed based on deep manifold learning to increase the efficiency of 3D nacelle design. The Pareto front curves under drag divergence constraints reveal the correlation between the geometry distribution and the surface isentropic Mach number distribution of 3D nacelles. This paper demonstrates the feasibility of the proposed geometric dimensionality reduction method for direct multi-objective optimization of 3D nacelles.

Spines and surgery descriptions of graph manifolds

Graph manifolds are compact orientable 3–manifolds obtained by gluing several copies of D2×S1 and N2×S1 together by homeomorphisms of some components of their boundaries (D2 is the 2–disc and N2 denotes the 2–disc with two holes). Here we study spines and surgery representations of orientable graph manifolds, and derive geometric presentations of their fundamental group. Then we determine the homeomorphism type of many Takahashi manifolds and the Teragaito manifolds, showing that they are graph manifolds with specified invariants. Finally, we describe graph manifolds arising from toroidal surgeries on certain classes of hyperbolic knots.

Unraveling the highly nonlinear dynamics of KCN molecular system using Lagrangian descriptors

In this work, we identify the phase-space structures which are responsible for the chaotic dynamics observed in KCN molecular system using the Lagrangian descriptors. We show that the vibrational dynamics of this molecule is strongly determined by the invariant manifolds associated with a particular stretching periodic orbit previously described (Párraga et al., 2018). Likewise, the representation of these invariant manifolds on a Poincaré surface of section is also studied, concluding that the intricate depiction that is observed has its origin in the complex behavior of the manifolds, which is a consequence of the strong anharmonicities in the potential energy surface.

Spectral analysis of causal dynamical triangulations via finite element method

We examine the dual graph representation of simplicial manifolds in causal dynamical triangulations (CDT) as a means to build observables and propose a new representation based on the finite element methods (FEM). In particular, with the application of FEM techniques, we extract the (low-lying) spectrum of the Laplace-Beltrami (LB) operator on the Sobolev space ${H}^{1}$ of scalar functions on piecewise flat manifolds and compare them with corresponding results obtained by using the dual graph representation. We show that, except for nonpathological cases in two dimensions, the dual graph spectrum and spectral dimension do not generally agree, neither quantitatively nor qualitatively, with the ones obtained from the LB operator on the continuous space. We analyze the reasons for this discrepancy and discuss its possible implications on the definition of generic observables built from the dual graph representation.

Computational mean-field games on manifolds

Conventional Mean-field games/control study the behavior of a large number of rational agents moving in Euclidean spaces. In this work, we explore the mean-field games on Riemannian manifolds. We formulate the mean-field game Nash Equilibrium on manifolds. We also establish the equivalence between the PDE system and the optimality conditions of the associated variational form on manifolds. Based on the triangular mesh representation of two-dimensional manifolds, we design a proximal gradient method for variational mean-field games. Our comprehensive numerical experiments on various manifolds illustrate the effectiveness and flexibility of the proposed model and numerical methods.

Representation of Manifolds for the Stochastic Swift-Hohenberg Equation with Multiplicative Noise

Representation of approximation for manifolds of the stochastic Swift-Hohenberg equation with multiplicative noise has been investigated via non-Markovian reduced system. The approximate parameterizations of the small scales for the large scales are given in the process of seeking for stochastic parameterizing manifolds, which are obtained as pullback limits of some backward-forward systems depending on the time-history of the dynamics of the low modes in a mean square sense through the nonlinear terms. When the corresponding pullback limits of some backward-forward systems are efficiently determined, the corresponding non-Markovian reduced systems can be obtained for researching good modeling performances in practice.

A Registration Method for Model Order Reduction: Data Compression and Geometry Reduction

We propose a general --- i.e., independent of the underlying equation --- registration method for parameterized Model Order Reduction. Given the spatial domain $\Omega \subset \mathbb{R}^d$ and a set of snapshots $\{ u^k \}_{k=1}^{n_{\rm train}}$ over $\Omega$ associated with $n_{\rm train}$ values of the model parameters $\mu^1,\ldots, \mu^{n_{\rm train}} \in \mathcal{P}$, the algorithm returns a parameter-dependent bijective mapping $\boldsymbol{\Phi}: \Omega \times \mathcal{P} \to \mathbb{R}^d$: the mapping is designed to make the mapped manifold $\{ u_{\mu} \circ \boldsymbol{\Phi}_{\mu}: \, \mu \in \mathcal{P} \}$ more suited for linear compression methods. We apply the registration procedure, in combination with a linear compression method, to devise low-dimensional representations of solution manifolds with slowly-decaying Kolmogorov $N$-widths; we also consider the application to problems in parameterized geometries. We present a theoretical result to show the mathematical rigor of the registration procedure. We further present numerical results for several two-dimensional problems, to empirically demonstrate the effectivity of our proposal.

Distance difference representations of Riemannian manifolds

Let M be a complete Riemannian manifold and $$F\subset M$$ a set with a nonempty interior. For every $$x\in M$$ , let $$D_x$$ denote the function on $$F\times F$$ defined by $$D_x(y,z)=d(x,y)-d(x,z)$$ where d is the geodesic distance in M. The map $$x\mapsto D_x$$ from M to the space of continuous functions on $$F\times F$$ , denoted by $${\mathcal {D}}_F$$ , is called a distance difference representation of M. This representation, introduced recently by Lassas and Saksala, is motivated by geophysical imaging among other things. We prove that the distance difference representation $${\mathcal {D}}_F$$ is a locally bi-Lipschitz homeomorphism onto its image $${\mathcal {D}}_F(M)$$ and that for every open set $$U\subset M$$ the set $${\mathcal {D}}_F(U)$$ uniquely determines the Riemannian metric on U. Furthermore the determination of M from $${\mathcal {D}}_F(M)$$ is stable if M has a priori bounds on its diameter, curvature, and injectivity radius. This extends and strengthens earlier results by Lassas and Saksala.

Efficient representation of invariant manifolds of periodic orbits in the CRTBP

<p style='text-indent:20px;'>This paper deals with a methodology for defining and computing an analytical Fourier-Taylor series parameterisation of local invariant manifolds associated to periodic orbits of polynomial vector fields. Following the Parameterisation Method, the functions involved in the series result by solving some linear non autonomous differential equations. Exploiting the Floquet normal form decomposition, the time dependency is removed and the differential problem is rephrased as an algebraic system to be solved for the Fourier coefficients of the unknown periodic functions. The procedure leads to an efficient and fast computational algorithm. Motivated by mission design purposes, the technique is applied in the framework of the Circular Restricted Three Body problem and the parameterisation of local invariant manifolds for several halo orbits is computed and discussed.

Methods of Bifurcation Theory in Multiparameter Problems of Hydroaeroelasticity

Applying bifurcation theory methods to nonlinear boundary value problems for ordinary differential equations (ODEs) of the fourth and higher orders encounters technical difficulties related to studying the spectrum of direct and adjoint linearized problems and constructing Green functions (i.e., proving the spectral problems to be Fredholm and determining the manifolds of bifurcation points). In order to overcome these difficulties, methods for separating the roots of relevant characteristic equations have been proposed, with the subsequent representation of the bifurcation manifolds in terms of these roots; this enables investigation of nonlinear problems in their rigorous statement. Such an approach is considered using the example of a two-point boundary value problem for a fourth-order ODE in which a statically bent pipeline section is described as a flexible elastic hollow rod, with a liquid flowing inside it, that is compressed or stretched by external boundary conditions with a free sliding left end and fixedly mounted right ends.

Affine representations of Fibonacci groups and flat manifolds

ABSTRACTWe construct examples of affine representations for a family of Fibonacci groups. As an application, we prove that all Hantzsche–Wendt groups of the same dimension are epimorphic images of the same Fibonacci group.

How to Depict 5-Dimensional Manifolds

We usually think of 2-dimensional manifolds as surfaces embedded in Euclidean 3-space. Since humans cannot visualise Euclidean spaces of higher dimensions, it appears to be impossible to give pictorial representations of higher-dimensional manifolds. However, one can in fact encode the topology of a surface in a 1-dimensional picture. By analogy, one can draw 2-dimensional pictures of 3-manifolds (Heegaard diagrams), and 3-dimensional pictures of 4-manifolds (Kirby diagrams). With the help of open books one can likewise represent at least some 5-manifolds by 3-dimensional diagrams, and contact geometry can be used to reduce these to drawings in the 2-plane. In this paper, I shall explain how to draw such pictures and how to use them for answering topological and geometric questions. The work on 5-manifolds is joint with Fan Ding and Otto van Koert.

Fock Representations and Deformation Quantization of Kähler Manifolds

The goal of this paper is to construct the Fock representation of noncommutative Kahler manifolds. Noncommutative Kahler manifolds studied here are constructed by deformation quantization with separation of variables, which was given by Karabegov. The algebra of the noncommutative Kahler manifolds contains the Heisenberg-like algebras. Local complex coordinates and partial derivatives of a Kahler potential satisfy the commutation relations between creation and annihilation operators. A Fock space is constituted by states obtained by applying creation operators on a vacuum which is annihilated by all annihilation operators. The algebras on noncommutative Kahler manifolds are represented as those of linear operators acting on the Fock space. In representations studied here, creation operators and annihilation operators are not Hermitian conjugates of one other, in general. Therefore, the basis vectors of the Fock space are not the Hermitian conjugates of those of the dual vector space. In this sense, we call the representation the twisted Fock representation. In this presentation, we construct the twisted Fock representations for arbitrary noncommutative Kahler manifolds given by deformation quantization with separation of variables, and give a dictionary to translate between the twisted Fock representations and functions on noncommutative Kahler manifolds concretely.

Twisted Fock representations of noncommutative Kähler manifolds

We introduce twisted Fock representations of noncommutative Kahler manifolds and give their explicit expressions. The twisted Fock representation is a representation of the Heisenberg like algebra whose states are constructed by applying creation operators to a vacuum state. “Twisted” means that creation operators are not Hermitian conjugate of annihilation operators in this representation. In deformation quantization of Kahler manifolds with separation of variables formulated by Karabegov, local complex coordinates and partial derivatives of the Kahler potential with respect to coordinates satisfy the commutation relations between the creation and annihilation operators. Based on these relations, we construct the twisted Fock representation of noncommutative Kahler manifolds and give a dictionary to translate between the twisted Fock representations and functions on noncommutative Kahler manifolds concretely.

Three-dimensional representations of the tube manifolds of the planar restricted three-body problem

The stable and unstable manifolds of the Lyapunov orbits of the Lagrangian equilibrium points L1, L2 play a key role in the understanding of the complicated dynamics of the circular restricted three-body problem. By developing a recent technique of computation of the stable and unstable manifolds, based on the use of Fast Lyapunov Indicators modified by the introduction of a filtering window function, we compute sample three-dimensional representations of the manifolds which show an original vista about their complicated development in the phase-space.

Constructing reference metrics on multicube representations of arbitrary manifolds

Reference metrics are used to define the differential structure on multicube representations of manifolds, i.e., they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous. This paper introduces a general procedure for constructing reference metrics automatically on multicube representations of manifolds with arbitrary topologies. The method is tested here by constructing reference metrics for compact, orientable two-dimensional manifolds with genera between zero and five. These metrics are shown to satisfy the Gauss–Bonnet identity numerically to the level of truncation error (which converges toward zero as the numerical resolution is increased). These reference metrics can be made smoother and more uniform by evolving them with Ricci flow. This smoothing procedure is tested on the two-dimensional reference metrics constructed here. These smoothing evolutions (using volume-normalized Ricci flow with DeTurck gauge fixing) are all shown to produce reference metrics with constant scalar curvatures (at the level of numerical truncation error).

Tight Graph Framelets for Sparse Diffusion MRI q-Space Representation.

In diffusion MRI, the outcome of estimation problems can often be improved by taking into account the correlation of diffusion-weighted images scanned with neighboring wavevectors in q-space. For this purpose, we propose in this paper to employ tight wavelet frames constructed on non-flat domains for multi-scale sparse representation of diffusion signals. This representation is well suited for signals sampled regularly or irregularly, such as on a grid or on multiple shells, in q-space. Using spectral graph theory, the frames are constructed based on quasi-affine systems (i.e., generalized dilations and shifts of a finite collection of wavelet functions) defined on graphs, which can be seen as a discrete representation of manifolds. The associated wavelet analysis and synthesis transforms can be computed efficiently and accurately without the need for explicit eigen-decomposition of the graph Laplacian, allowing scalability to very large problems. We demonstrate the effectiveness of this representation, generated using what we call tight graph framelets, in two specific applications: denoising and super-resolution in q-space using ℓ0 regularization. The associated optimization problem involves only thresholding and solving a trivial inverse problem in an iterative manner. The effectiveness of graph framelets is confirmed via evaluation using synthetic data with noncentral chi noise and real data with repeated scans.