Maps Of Manifolds Research Articles

Theory of Holomorphic Maps of Two-Dimensional Complex Manifolds to Toric Manifolds and Type-A Multi-String Theory

We study the field theory localizing to holomorphic maps from a complex manifold of complex dimension 2 to a toric target (a generalization of A model). Fields are realized as maps to {{(mathbb{C}text{*})}^{N}} where one includes special observables supported on (1,1)-dimensional submanifolds to produce maps to the toric compactification. We study the mirror of this model. It turns out to be a free theory interacting with {{N}_{{{text{comp}}}}} topological strings of type A. Here, {{N}_{{{text{comp}}}}} is the number of compactifying divisors of the toric target. Before the mirror transformation, these strings are vortex (actually, holomortex) strings.

Conformality: classical and generalized

We discuss several topics: the concept of conformal mapping of Riemannian and pseudo-Riemannian manifolds, conformal rigidity of higher-dimensional domains, and conformal flexibility of two-dimensional domains of Euclidian and Minkowski planes. We present an extension of the concept of conformal mapping proposed by M. Gromov and recall an open problem related to it.

Multifidelity modeling similarity conditions for airfoil dynamic stall prediction with manifold mapping

PurposeThe purpose of this work is to investigate the similarity requirements for the application of multifidelity modeling (MFM) for the prediction of airfoil dynamic stall using computational fluid dynamics (CFD) simulations.Design/methodology/approachDynamic stall is modeled using the unsteady Reynolds-averaged Navier–Stokes equations and Menter's shear stress transport turbulence model. Multifidelity models are created by varying the spatial and temporal discretizations. The effectiveness of the MFM method depends on the similarity between the high- (HF) and low-fidelity (LF) models. Their similarity is tested by computing the prediction error with respect to the HF model evaluations. The proposed approach is demonstrated on three airfoil shapes under deep dynamic stall at a Mach number 0.1 and Reynolds number 135,000.FindingsThe results show that varying the trust-region (TR) radius (λ) significantly affects the prediction accuracy of the MFM. The HF and LF simulation models hold similarity within small (λ ≤ 0.12) to medium (0.12 ≤ λ ≤ 0.23) TR radii producing a prediction error less than 5%, whereas for large TR radii (0.23 ≤ λ ≤ 0.41), the similarity is strongly affected by the time discretization and minimally by the spatial discretization.Originality/valueThe findings of this work present new knowledge for the construction of accurate MFMs for dynamic stall performance prediction using LF model spatial- and temporal discretization setup and the TR radius size. The approach used in this work is general and can be used for other unsteady applications involving CFD-based MFM and optimization.

Improved Well-Control Progressive Optimization with Generalized Barycentric Coordinates and Manifold Mapping

Summary Well-control management is nowadays frequently approached by means of mathematical optimization. However, in many practical situations the optimization algorithms used are still computationally expensive. In this paper, we present progressive optimization (PO), a simulator-nonintrusive four-stage methodology to accelerate optimal search substantially in well-control applications. The first stage of PO comprises a global exploration of the search space using design of experiments (DOEs). Thereafter, in the second stage, a fast-to-evaluate proxy model is constructed with the points considered in the experimental design. This proxy is based on generalized barycentric coordinates (GBCs), a generalization of the concept of barycentric coordinates used within a triangle. GBCs can be especially suited to problems in which nonlinearities are not strong, as is the case often for well-control optimization. This fact is supported by the good performance in these types of optimization problems of techniques that rely strongly on linearity assumptions, such as trajectory piecewise linearization, a procedure that is not always applicable due to its simulator-intrusive nature. In the third stage, the precision of the proxy model is iteratively improved and the enhanced surrogate model is reoptimized by means of manifold mapping (MM), a method that combines models with different levels of accuracy. MM has solid theoretical foundations and leads to efficient optimization schemes in multiple engineering disciplines. The final and fourth stage aims at additional improvement, resorting to direct optimization of the best solution from the previous stages. Nonlinear (operational) constraints are handled in PO with the filter method. The optimal search may be finalized earlier than at the fourth stage whenever the solution obtained is of satisfactory quality. PO is tested on two waterflooding problems built upon a synthetic model previously studied in well-control optimization literature. In these problems, which have 120 and 40 well controls and include nonlinear constraints, we observe for PO reductions in computational cost, for solutions of comparable quality, of approximately 30% and 50% with respect to Hooke-Jeeves direct search (HJDS), which, in turn, outperforms particle swarm optimization (PSO). HJDS and PSO are simulator-nonintrusive algorithms that usually perform well in optimization for oilfield operations. The novel concepts of GBC and MM within the framework of the PO paradigm can be extremely helpful for practitioners to efficiently deal with optimized well-control management. Savings of 50% in computing cost may be translated in practice into days of computations for just a single field and optimization run.

Conformality, classical and generalized

We discuss several topics: the concept of conformal mapping of Riemannian and pseudo-Riemannian manifolds, conformal rigidity of higher-dimensional domains, and conformal flexibility of two-dimensional domains of Euclidian and Minkowski planes. We present an extension of the concept of conformal mapping proposed by M. Gromov and recall an open problem related to it.

Data-driven fault diagnosis for wind turbines using modified multiscale fluctuation dispersion entropy and cosine pairwise-constrained supervised manifold mapping

Condition monitoring and fault diagnosis of wind turbines is an attractive yet challenging task. This paper presents a novel data-driven fault diagnosis scheme for wind turbines, using refined time-shift multiscale fluctuation-based dispersion entropy (RTSMFDE) and cosine pairwise-constrained supervised manifold mapping (CPCSMM). The developed RTSMFDE can improve multiscale dispersion entropy in complexity measurement of time series, by employing the refined time-shift analysis approach. Moreover, the proposed CPCSMM combines cosine distance metric, supervised theory and sparse global manifold structure, to obtain the most important features, resulting in a low dimensional representation. In this fault diagnosis scheme, a high-dimensional fault feature set is first constructed using RTSMFDE. Then, sensitive features are extracted using CPCSMM. Finally, the low-dimensional feature set is fed to a beetle antennae search-based support vector machine for fault identification. Simulation experiments and wind turbine fault diagnosis experiments were used, to show that the proposed RTSMFDE can comprehensively mine the wind turbine features and outperforms the existing state-of-the-art entropy methods. The visualization results and classification effects of the proposed CPCSMM approach are better than existing unsupervised and supervised dimensionality reduction methods. The proposed data-driven fault diagnosis method can effectively and accurately identify multi-faults and outperforms existing fault diagnosis techniques.

Extension of the curvature form of the relative canonical line bundle on families of Calabi–Yau manifolds and applications

Given a proper, open, holomorphic map of Kahler manifolds, whose general fibers are Calabi-Yau manifolds, the volume forms for the Ricci-flat metrics induce a hermitian metric on the relative canonical bundle over the regular locus of the family. We show that the curvature form extends as a closed positive current. Consequently the Weil-Petersson metric extends as a positive current. In the projective case, the Weil-Petersson form is known to be the curvature of a certain determinant line bundle, equipped with a Quillen metric. As an application we get that after blowing up the singular locus, the determinant line bundle extends, and the Quillen metric extends as singular hermitian metric, whose curvature is a positive current.

Single- and Multipoint Aerodynamic Shape Optimization Using Multifidelity Models and Manifold Mapping

In this paper, a computationally efficient multifidelity local search algorithm for aerodynamic design optimization is presented. In this paper’s approach, direct optimization of a computationally expensive model is replaced by an iterative updating and reoptimization of a fast multifidelity model constructed using a low-fidelity model adapted locally using manifold mapping (MM) to become a reliable representation of the high-fidelity one during the optimization process. Only one high-fidelity model evaluation is needed per design iteration, and no gradient information is used. The proposed method is validated and characterized by applying it to a few single- and multipoint optimization problems involving inviscid and viscous transonic flows. The proposed method is compared with the sequential least-squares programming (SLSQP) gradient-based approach with the gradients calculated based on adjoint sensitivities. In the inviscid single-point test case, a drag reduction of 410.8 counts was achieved by the MM algorithm while requiring approximately 1469 min on a high performance computing (HPC) with 32 processors. SLSQP with adjoints achieved a drag reduction of 425.9 counts, while requiring approximately 1536 min under the same HPC setup. For the viscous single-point test case, a 83.2 drag count reduction was reported for the MM compared to 83.4 for SLSQP with adjoints. In that case, the MM algorithm was around eight times faster in terms of computing time. In the multipoint design test case, the MM algorithm was computationally cheaper by at least an order of magnitude compared to SLSQP with adjoints, although the objective function value was around three drag counts higher. Furthermore, it was found that in the multipoint cases the MM algorithm scales favorably with the number of operational conditions considered compared to SLSQP with adjoints.

A Digital Twin-Driven Method for Product Performance Evaluation Based on Intelligent Psycho-Physiological Analysis

Abstract Digital twin, a new emerging and fast-growing technology which is one of the most promising technologies for smart design and manufacturing, has attracted much attention worldwide recently. With the application of digital twin, product performance evaluation has entered the data-driven era. However, traditional methods for evaluation mainly place emphasis on structure analysis in the stage of manufacturing and service in digital twin. They cannot synthesize multi-source information and take the high-level emotional response into consideration in the design stage. To overcome these disadvantages, a digital twin-driven method is proposed evaluating product design schemes in this study. It enables the acquisition of electroencephalogram (EEG) data, physical data, and emotional feedback. Human factors are systematically considered in the evaluation process to establish the information association between EEG and performance levels. Moreover, intelligent psycho-physiological analysis that incorporates EEG into the fuzzy comprehensive evaluation (FCE) and machine learning methods is adopted within the proposed method. It synthesizes human factors such as psychological requirements, subjective and objective assessment indicators to realize a novel machine learning-based EEG analysis. Taking advantage of the binary particle swarm optimization (BPSO) improved Riemannian manifold mapping, Riemann geometry (RG) features are extracted and selected from EEG signals. Differences of implicit psychological states while using the product produced by different design schemes can be more easily detected and classified. A case study of high-speed elevator is conducted to verify the feasibility and effectiveness of the proposed method. The accuracy of EEG classification for performance evaluation reaches 92%.

Cartography as Spatial Representation: A new assessment of the competing advantages and drawbacks across fields of science

The history of cartography has been marked by the endless search for the perfect form for the representation of the information on a spherical surface manifold into the flat planar format of the printed page or computer screen. Dozens of cartographic formats have been proposed over the centuries from ancient Greek times to the present. This is an issue not just for the mapping of the globe, but in all fields of science where spherical entities are found. The perceptual and representational advantages and drawbacks of many of these formats are considered, particularly in the tension between a unified representation, which is always distorted in some dimension, and a minimally distorted representation, which can only be obtained by segmentation into sectorial patches. The use of these same formats for the mapping of spherical manifolds are evaluated, from quantum physics through the mapping of the brain to the large-scale representation of the cosmos.

Adaptive Flexible Optimal Graph for Unsupervised Dimensionality Reduction

Graph-based dimensionality reduction methods are widely used in classification and clustering tasks due to their superior performance. The key to the performance of these methods is how to construct the graph. Although some existing methods can obtain an adaptive graph and a projection matrix simultaneously by combining manifold learning and graph construction in a unified framework, the linear constraint used in manifold learning is too hard . To solve this problem, we propose a novel method named adaptive flexible optimal graph (AFOG) for unsupervised dimensionality reduction. AFOG can obtain an adaptive flexible optimal graph by relaxing the linear constraint in the process of low-dimensional manifold mapping. At the same time, by using the principle of maximum separability, it can also obtain an effective projection matrix, which can solve out-ofsample problems. Experiments on six public benchmark data sets indicate that AFOG outperforms several other state-of-the-art methods.

Multifidelity flutter prediction using local corrections to the generalized AIC

This work adopts the manifold mapping multifidelity modeling technique for frequency-domain flutter prediction. While this type of multifidelity model is usually employed in a trust-region optimization framework, the current approach uses it, in combination with the p-k method, to find a critical point in the high-fidelity Mach-reduced frequency space (namely, the matched flutter point for a given density and sound speed). This is done by first computing a grid of low-fidelity (doublet lattice-based) aerodynamic influence coefficients in modal coordinates. After starting at the low-fidelity match point, the process iteratively approaches the high-fidelity (Euler-based) match point using a single Mach and reduced frequency evaluation per iteration. The locally accurate multifidelity model operates by applying a correction matrix to the low-fidelity response using the changes in high- and low-fidelity response vectors between iterations; the correction matrix is computed using the Moore-Penrose pseudoinverse operator, making it reminiscent of various doublet lattice correction methods. When applied to a common test case, the process is shown to converge in 4 to 8 high-fidelity evaluations, depending on the fluid density and sound speed. For comparison, standard time- and frequency domain methods are estimated to be 3 to 10 times more costly in terms of time steps or frequency evaluations.

Manifold Sparse Auto-Encoder for Machine Fault Diagnosis

Although the use of deep learning algorithms to find effective features for fault diagnosis has somewhat enhanced of fault classification accuracy, the lack of guidelines and the parameters such as layers of the deep learning architecture and dimension of each hidden-level has limited further improvement. Based on manifold mapping eigenvalues, an optimized deep learning model, called manifold sparse auto-encoder (MSAE) neural network, is constructed to diagnose the machine faults. Two main contributions of this paper can be summarized as (1) every encoding and decoding process is taken as a module to decline the vanishing gradient problem, and (2) the dimension of each hidden layer is determined by the manifold mapping eigenvalues of hidden neurones, whereas the layers of the deep learning architecture are determined by the clustering distribution of features. Gearbox datasets demonstrated that the proposed MSAE can extract better discriminative high-level features and has higher accuracy in machinery fault diagnosis compared with other machine learning methods.

Multiplicity results involving p-biharmonic Kirchhoff-type problems

This paper deals with the existence of multiple solutions for the following Kirchhoff type equations involving p-biharmonic operator:−M(∫Ω(|Δpu|2+|u|p)dx)(Δp2u−|u|p−2u)=λf(x)|u|q−2u+g(x)|u|m−2u,x∈Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& -M \\biggl( \\int_{\\varOmega} \\bigl( \\vert \\Delta_{p}u \\vert ^{2}+ \\vert u \\vert ^{p} \\bigr)\\,dx \\biggr) \\bigl( \\Delta _{p}^{2}u- \\vert u \\vert ^{p-2}u \\bigr) =\\lambda f(x) \\vert u \\vert ^{q-2}u+g(x) \\vert u \\vert ^{m-2}u,\\quad x\\in\\varOmega, \\end{aligned}$$ \\end{document} where Ω is a bounded domain in mathbb{R}^{N} (N>1), lambda >0, p, q, m>1, M is a continuous function, and the weight functions f and g are measurable. We obtain the existence results by combining the variational method with Nehari manifold and fibering maps.

Maps with Prescribed Boardman Singularities

In this paper we extend Y. Eliashberg’s theorem on the maps with fold type singularities to arbitrary Thom-Boardman singularities. Namely, we state a necessary and sufficient condition for a continuous map of smooth manifolds of the same dimension to be homotopic to a generic map with a prescribed Thom-Boardman singularity ΣI at each point. In dimensions 2 and 3 we rephrase this condition in terms of the homology classes of the given singular loci and the characteristic classes of the manifolds.

On the Equicontinuity of Families of Inverse Mappings of Riemannian Manifolds

We study homeomorphisms of Riemannian manifolds with unbounded characteristic such that the inverse mappings satisfy the Poletsky-type inequality. It is established that their families are equicontinuous if the function Q which is related to the Poletsky inequality and is responsible for a distortion of the modulus, is integrable in the given domain, here the original manifold is connected and the domain of definition and the range of values of mappings have compact closures.

On the equicontinuity of families of inverse mappings of Riemannian manifolds

We study homeomorphisms of Riemannian manifolds with unbounded characteristic such that the inverse mappings satisfy the Poletsky-type inequality. It is established that their families are equicontinuous if the function Q which is related to the Poletsky inequality and is responsible for a distortion of the modulus, is integrable in the given domain, here the original manifold is connected and the domain of definition and the range of values of mappings have compact closures.

On the Local Behavior of Sobolev Classes on Two-Dimensional Riemannian Manifolds

We study open discrete maps of two-dimensional Riemannian manifolds from the Sobolev class. For these mappings, we establish the lower estimates of distortions of the moduli of families of the curves. As a consequence, we establish the equicontinuity of Sobolev classes at the interior points of the domain.

Geodesic Mappings of Vn(K)-Spaces and Concircular Vector Fields

In the present paper, we study geodesic mappings of special pseudo-Riemannian manifolds called V n ( K ) -spaces. We prove that the set of solutions of the system of equations of geodesic mappings on V n ( K ) -spaces forms a special Jordan algebra and the set of solutions generated by concircular fields is an ideal of this algebra. We show that pseudo-Riemannian manifolds admitting a concircular field of the basic type form the class of manifolds closed with respect to the geodesic mappings.

Existence and multiplicity of solutions for fractional Laplacian system

ABSTRACT This paper is devoted to study the existence and multiplicity of solutions to fractional -Laplacian system by analyzing the relationship between the Nehari manifold and fibering maps.