Manifold Form Research Articles

A novel and fully automated coordinate system transformation scheme for near optimal surrogate construction

This work develops a novel coordinate system transformation scheme to improve the performance of common radial basis function surrogate models. This coordinate system transformation scheme is based on the fact that commonly used basis functions are isotropic.Three main empirical findings are established in this study. Firstly, in general isotropic functions are inadequate to describe anisotropic data manifolds due to a mismatch between the functional form and the form of the data manifold resulting in poor generative performance. Counter-intuitively, utilising additional gradients during surrogate training often worsens the generative capability.Secondly, component-wise scaling of isotropic model forms during cross-validation is inadequate to enhance the functional form of the data manifold form as anisotropic coupling in the data manifold remains coupled. Improving the match between the functional form and the data manifold form requires both rotation and scaling.Thirdly, the coordinate system transformation scheme should predominantly be based on a collection of local curvature estimations and not on global curvature approximations. Gradients are critical to estimating the local curvature for identifying a near-optimal reference frame for surrogate construction, which then translates to additional benefits of gradients in gradient-enhanced surrogates.Based on the above observations, this paper proposes an isotropic transformation for the data coordinate system that performs near-optimal transformations on lower dimensional data without requiring any cross-validation. The method is compared against commonly applied component-wise cross-validation data coordinate system scaling as well as the more modern Active Subspace Method on a carefully crafted decomposable test problem, which has a known optimal coordinate system, that varies between 2 and 16 dimensions.The paper concludes after demonstrating that the developed transformation scheme, as well as the other common methods, will offer little benefit on non-decompose problems and offers some suggestions on future work to create a more general isotropic transformation.

On double phase Kirchhoff problems with singular nonlinearity

AbstractIn this paper, we study multiplicity results for double phase problems of Kirchhoff type with right-hand sides that include a parametric singular term and a nonlinear term of subcritical growth. Under very general assumptions on the data, we prove the existence of at least two weak solutions that have different energy sign. Our treatment is based on the fibering method in form of the Nehari manifold. We point out that we cover both the nondegenerate as well as the degenerate Kirchhoff case in our setting.

Efficient identification of Alzheimer’s brain dynamics with Spatial-Temporal Autoencoder: A deep learning approach for diagnosing brain disorders

Alzheimer’s disease (AD) is a progressive neurological disorder seriously affecting cognitive and behavior abilities of the older people. Accurate and early diagnosis of AD is critical for improving the therapeutic effect and alleviate the clinical symptom. In this work, we proposed an automatic EEG-based diagnosis scheme for AD patients with deep learning methods. A Spatial-Temporal Autoencoder (STAE) framework with Convolutional Neural Network (CNN)- Long Short-Term-Memory (LSTM) generative model was designed to estimate latent factors of the observed oscillatory activity in the brain from multi-channels electroencephalogram (EEG) signals via unsupervised learning. Based on latent factor analysis, Alzheimer’s brain dynamics on single-trials were inferred and temporal evolution of latent brain state was analyzed in low-dimensional state space. The study mainly showed that: i) the brain state trajectories of AD patients were distinct from healthy subjects, resulting in different forms of ring manifolds and allowing to accurately identify AD; ii) experimental results demonstrated the efficiency and flexibility of the proposed deep learning-based diagnosis scheme, by which the classification of AD patients and the normal based on clinical EEG dataset achieved an accuracy of 96.30%, a sensitivity of 97.73% and a specificity of 94.69% with a multiple layer perception (MLP) classifier; iii) compared with other approaches for latent brain dynamics estimation, STAE exhibited superior performance with high accuracy of AD identification and strongly robust against instabilities in EEG recordings. The present results reveal the neuropathological mechanism of Alzheimer’s disease with brain dynamics variations and provide a feasible diagnosis tool for brain disorders.

Curvature-Adaptive Meta-Learning for Fast Adaptation to Manifold Data.

Meta-learning methods are shown to be effective in quickly adapting a model to novel tasks. Most existing meta-learning methods represent data and carry out fast adaptation in euclidean space. In fact, data of real-world applications usually resides in complex and various Riemannian manifolds. In this paper, we propose a curvature-adaptive meta-learning method that achieves fast adaptation to manifold data by producing suitable curvature. Specifically, we represent data in the product manifold of multiple constant curvature spaces and build a product manifold neural network as the base-learner. In this way, our method is capable of encoding complex manifold data into discriminative and generic representations. Then, we introduce curvature generation and curvature updating schemes, through which suitable product manifolds for various forms of data manifolds are constructed via few optimization steps. The curvature generation scheme identifies task-specific curvature initialization, leading to a shorter optimization trajectory. The curvature updating scheme automatically produces appropriate learning rate and search direction for curvature, making a faster and more adaptive optimization paradigm compared to hand-designed optimization schemes. We evaluate our method on a broad set of problems including few-shot classification, few-shot regression, and reinforcement learning tasks. Experimental results show that our method achieves substantial improvements as compared to meta-learning methods ignoring the geometry of the underlying space.

ПЕРСПЕКТИВИ ОЦІНЮВАННЯ ТРИВИМІРНИХ ДЕФОРМАЦІЙ ЗЕМЛІ ЗА ДАНИМИ ГЛОБАЛЬНИХ НАВІГАЦІЙНИХ СУПУТНИКОВИХ СИСТЕМ

Presented in the article information is of a scientific and methodical character. Accumulation of coordinate monitoring data in networks of permanent GNSS stations made it possible to increase the efficiency of solving of many problems of moderngeodynamics. At the same time, their use made it necessary to rethink traditional theoretical approaches to solving of the problem, in particular, in terms of developing methods for evaluation of three-dimensional deformations.The analysis of the used theoretical approaches and methods for evaluation of three-dimensional deformations of the Earth in this article are presented. The insufficient degree of their correspondence to the potential information resource of GNSS data is substantiated.The method for evaluation of deformations of the Earth from the standpoint of the theory of transformations of Riemannian space imagesin the form of complex diffeomorphic manifolds is proposed. Properties of complex manifolds in the form of a tangent Euclidean space, which is parameterized by a rectangular Cartesian coordinate system, are used. This choice is justi-fied by its identity with the ITRS system, which is used to monitoring of coordinates by the GNSS method. The algorithm for constructing a functional model of deformation based on the homeomorphism of transformations is formulated. The method involves the direct use of coordinates in the ITRS system and modeling of deformations of the curvilinear topographic surfaceof the Earth in three-dimensional space. The method provides evaluation of three-dimensional deformation tensors and related invariants, which belong to the topographic surface. Working formulas for evaluation of main linear deformations, dilatation and rigid rotation of the Earth have been obtained. The method is generalized for evaluation of nonlinear deformations

How is a graph not like a manifold?

For an equivariantly formal action of a compact torus $T$ on a smooth manifold $X$ with isolated fixed points we investigate the global homological properties of the graded poset $S(X)$ of face submanifolds. We prove that the condition of the $j$-independency of tangent weights at each fixed point implies the $(j+1)$-acyclicity of the skeleta $S(X)_r$ for $r>j+1$. This result provides a necessary topological condition for a GKM graph to be a GKM graph of some GKM manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold of dimension $2n$ with an $(n-1)$-independent action of the $(n-1)$-dimensional torus, under certain colourability assumptions on its GKM graph. This description relates the equivariant cohomology algebra to the face algebra of a simplicial poset. This observation underlines a certain similarity between actions of complexity $1$ and torus manifolds. Bibliography: 27 titles.

Effects of bio-oil - diesel blends: the performance and emissions of diesel engines with intake manifold variations

Alternative fuels such as vegetable oil, biodiesel, and bio-oil are potential alternatives for diesel engines. Bio-oil is one of the products of the decomposition process of chemical and organic matter by adding heating process without oxygen is called rapid pyrolysis. Bio-oil in the pyrolysis process is made from Brem waste. Brem is a typical food from the district of Madiun, Indonesia. This research is important because the switch from petroleum-sourced fuel to Bio-Oil oil cannot be automatically applied to diesel engines. Spray quality and combustion process, type and quality of fuel, and engine design and condition are some of the important parameters affecting diesel engine performance. Research data is very important because it can determine the intake manifold and the best bio-oil-diesel blends to be used in the operation of diesel engines with bio-oil-diesel blends. The research uses three variations of the intake manifold form. Variations of research with lamp loading of 650, 1300, 1950 and 2600 watts determine specific power and fuel consumption. Use an opacity smoke meter to determine the opacity of diesel engine exhaust gases. The fuel used is pure diesel fuel and Bio-Oil fuel B10, B15, and B20 resulting from the pyrolysis of Brem waste with a fuel volume of 25 ml for each research variable. The results of testing the intake manifold variation and the bio-oil-diesel blended are the intake manifold variation 1 with a fuel mixture variation of 20 % Bio-Oil - 80% pure diesel (B20), which has the best results both electric power, specific fuel consumption and opacity.

Song of Songs 1:1—Text and Paratext

Abstract The argument draws upon literary theory to revisit the two clauses that, traditionally, make up Song 1:1. (1) The title evaluates the work as the song-most of songs. I argue that the evaluation refers to the work’s manifold form of simulation—a literary work representing the speech of a dreamer, who speaks from both inside and outside the dream. (2) The scoring in MT, the rubric in LXX (Sinaiticus, Alexandrinus), ancient interpreters and modern all take the first words of the Song of Songs to be a heading, comprising title (שִׁיר הַשִּׁירִים) and attribution (אֲשֶׁר לִשְׁלֹמֹה). I argue that the clause marked and understood as an attribution may be the beginning of the character’s speech.

Conformally formal manifolds and the uniformly quasiregular non-ellipticity of [formula omitted

We show that the manifold (S2×S2)#(S2×S2) does not admit a non-constant non-injective uniformly quasiregular self-map. This answers a question of Martin, Mayer, and Peltonen, and provides the first example of a quasiregularly elliptic manifold which is not uniformly quasiregularly elliptic.To obtain the result, we introduce conformally formal manifolds, which are closed smooth n-manifolds M admitting a measurable conformal structure [g] for which the (n/k)-harmonic k-forms of the structure [g] form an algebra. This is a conformal counterpart to the existing study of geometrically formal manifolds. We show that, similarly as in the geometrically formal theory, the real cohomology ring H⁎(M;R) of a conformally formal n-manifold M admits an embedding of algebras Φ:H⁎(M;R)↪∧⁎Rn. We also show that uniformly quasiregularly elliptic manifolds M are conformally formal in a stronger sense, in which the wedge product is replaced with a conformally scaled Clifford product. For this stronger version of conformal formality, the image of Φ is closed under the Euclidean Clifford product of ∧⁎Rn, which in turn is impossible for M=(S2×S2)#(S2×S2).

Dynamics of Delayed Phytoplankton–Zooplankton System with Disease Spread Among Zooplankton

In this paper, we study the dynamics of phytoplankton–zooplankton system with delay, where delay means that releasing toxin for phytoplankton is not instantaneous. First, we prove the positivity and boundedness of solutions, discuss the Hopf bifurcation caused by delay. Furthermore, we study the property of Hopf bifurcation by center manifold and normal form. Then, we study the global existence of bifurcated periodic solution. Finally by simulation, we show the influence of delay, disease spread and recovery from infected to susceptible on the dynamics of phytoplankton–zooplankton system.

Probabilistic modelling of general noisy multi-manifold data sets

The intrinsic nature of noisy and complex data sets is often concealed in low-dimensional structures embedded in a higher dimensional space. Number of methodologies have been developed to extract and represent such structures in the form of manifolds (i.e. geometric structures that locally resemble continuously deformable intervals of Rj1). Usually a-priori knowledge of the manifold's intrinsic dimensionality is required. Additionally, their performance can often be hampered by the presence of a significant high-dimensional noise aligned along the low-dimensional core manifold. In real-world applications, the data can contain several low-dimensional structures of different dimensionalities. We propose a framework for dimensionality estimation and reconstruction of multiple noisy manifolds embedded in a noisy environment. To the best of our knowledge, this work represents the first attempt at detection and modelling of a set of coexisting general noisy manifolds by uniting two aspects of multi-manifold learning: the recovery and approximation of core noiseless manifolds and the construction of their probabilistic models. The easy-to-understand hyper-parameters can be manipulated to obtain an emerging picture of the multi-manifold structure of the data. We demonstrate the workings of the framework on two synthetic data sets, presenting challenging features for state-of-the-art techniques in Multi-Manifold learning. The first data set consists of multiple sampled noisy manifolds of different intrinsic dimensionalities, such as Möbius strip, toroid and spiral arm. The second one is a topologically complex set of three interlocked toroids. Given the absence of such unified methodologies in the literature, the comparison with existing techniques is organized along the two separate aspects of our approach mentioned above, namely manifold approximation and probabilistic modelling. The framework is then applied to a complex data set containing simulated gas volume particles from a particle simulation of a dwarf galaxy interacting with its host galaxy cluster. Detailed analysis of the recovered 1D and 2D manifolds can help us to understand the nature of Star Formation in such complex systems.

The Faddeev-LeVerrier algorithm and the Pfaffian

We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(nβ+1) where n is the size of the matrix and O(nβ) is the cost of multiplying n×n-matrices, β∈[2,2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.

Lightlike manifolds and Cartan geometries

Lightlike Cartan geometries are introduced as Cartan geometries modelled on the future lightlike cone in Lorentz-Minkowski spacetime. Then, we provide an approach to the study of lightlike manifolds from this point of view. It is stated that every lightlike Cartan geometry on a manifold N provides a lightlike metric h with radical distribution globally spanned by a vector field Z. For lightlike hypersurfaces of a Lorentz manifold, we give the condition that characterizes when the pull-back of the Levi-Civita connection form of the ambient manifold is a lightlike Cartan connection on such hypersurface. In the particular case that a lightlike hypersurface is properly totally umbilical, this construction essentially returns the original lightlike metric. From the intrinsic point of view, starting from a given lightlike manifold (N, h), we show a method to construct a family of ambient Lorentzian manifolds that realize (N, h) as a hypersurface. This method is inspired on the Feffermann-Graham ambient metric construction in conformal geometry and provides a lightlike Cartan geometry on the original manifold when (N, h) is generic.

Parallel spinors on Lorentzian Weyl spaces

Holonomy algebras of Lorentzian Weyl spin manifolds with weighted parallel spinors are found. For Lorentzian Weyl manifolds admitting recurrent null vector fields are introduced special local coordinates similar to Kundt and Walker ones. Using that, the local form of all Lorentzian Weyl spin manifolds with weighted parallel spinors is given. The Einstein-Weyl equation for the obtained Weyl structures is analyzed and examples of Einstein-Weyl spaces with weighted parallel spinors are given.

Research on three‐phase VSR segmented PI synergetic control strategy based on LCL filter

In view of the resonance phenomenon caused by LCL filter introduced by voltage pulse width modulation (PWM) rectifier as well as the inability of traditional PI adjustment to meet the needs of different load changes, this article proposed a segmented PI synergetic control strategy for the LCL filter rectifier based on the proportional-integral manifold form synergetic control strategy. First of all, a synergetic controller based on the proportional-integral manifold was constructed; secondly, particle swarm optimization was used to optimize PI parameters under different load conditions; thirdly, the optimized PI parameters were used to establish a segmented trigger database, taking load impedance as the trigger condition. The control strategy made full use of the LCL filter's ability to suppress higher harmonics and the PI adjustment is simple and fast under specific loads. The synergetic control theory has a lot of advantages, including chatter-free, simple control ideas, easy implementation, etc. Simulation experiments verified that the control strategy could effectively reduce the harmonic content, avoid resonance phenomena and improve system stability, and it was characterized by high control accuracy and simple structure.

Неголономные многообразия Кенмоцу, оснащенные обобщенной связностью Танаки — Вебстера

А non-holonomic Kenmotsu manifold equipped with a connection analogous to the generalized Tanaka — Webster connection, is consid­ered. The studied connection is obtained from the generalized Tanaka — Webster connection by replacing the first structural endomorphism by the second structural endomorphism. The obtained connection is also called in the work the generalized Tanaka — Webster connection. Unlike a Kenmotsu manifold, the structure form of a non-holonomic Kenmotsu manifold is not closed. The consequence of this single differ­ence is a significant discrepancy in the properties of such manifolds. For example, it is proved in the paper that the alternation of the Ricci-Schouten tensor of a non-holonomic Kenmotsu manifold, which is a transverse analogue of the Ricci tensor, is proportional to the external differential of the structural form. At the same time, in the classical case of a Kenmotsu manifold, the Ricci — Schouten tensor is a symmetric tensor. It is proved that a Tanaka — Webster connection is a metric connec­tion. It is also proved that from the fact that the alternation of the Ricci-Schouten tensor is proportional to the external differential of the structur­al form, the following statement holds: if a non-holonomic Kenmotsu manifold is an Einstein manifold with respect to the generalized Tanaka — Webster connection, then it is Ricci-flat with respect to the same con­nection.

Dynamics of a diffusive delayed competition and cooperation system

Abstract In this manuscript, we first consider the diffusive competition and cooperation system subject to Neumann boundary conditions without delay terms and get the conclusion that the unique positive constant equilibrium is locally asymptotically stable. Then, we study the diffusive delayed competition and cooperation system subject to Neumann boundary conditions, and the existence of Hopf bifurcation at the positive equilibrium is obtained by regarding delay term as the parameter. By the theory of center manifold and normal form, an algorithm for determining the direction and stability of Hopf bifurcation is derived. Finally, some numerical simulations and summarizations are carried out for illustrating the theoretical analytic results.

A 4+1 Formalism for the Evolving Stueckelberg-Horwitz-Piron Metric

We propose a field theory for the local metric in Stueckelberg–Horwitz–Piron (SHP) general relativity, a framework in which the evolution of classical four-dimensional (4D) worldlines xμτ (μ=0,1,2,3) is parameterized by an external time τ. Combining insights from SHP electrodynamics and the ADM formalism in general relativity, we generalize the notion of a 4D spacetime M to a formal manifold M5=M×R, representing an admixture of geometry (the diffeomorphism invariance of M) and dynamics (the system evolution of Mτ with the monotonic advance of τ∈R). Strategically breaking the formal 5D symmetry of a metric gαβ(x,τ) (α,β=0,1,2,3,5) posed on M5, we obtain ten unconstrained Einstein equations for the τ-evolution of the 4D metric γμν(x,τ) and five constraints that are to be satisfied by the initial conditions. The resulting theory differs from five-dimensional (5D) gravitation, much as SHP U(1) gauge theory differs from 5D electrodynamics.

Gaussians on Riemannian Manifolds: Applications for Robot Learning and Adaptive Control

This article presents an overview of robot learning and adaptive control applications that can benefit from a joint use of Riemannian geometry and probabilistic representations. The roles of Riemannian manifolds, geodesics, and parallel transport in robotics are discussed, and several forms of manifolds already employed in robotics are explained. A varied range of techniques employing Gaussian distributions on Riemannian manifolds is then introduced, and two example applications are presented, involving the control of a prosthetic hand from surface electromyography (sEMG) data and the teleoperation of a bimanual underwater robot.

Manifolds of Isospectral Matrices and Hessenberg Varieties

Abstract We consider the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly formal manifold. The equivariant and ordinary cohomology rings of $X_h$ are described using GKM theory. The main goal of this paper is to show the connection between the manifolds $X_h$ and regular semisimple Hessenberg varieties well known in algebraic geometry. Both spaces $X_h$ and Hessenberg varieties form wonderful families of submanifolds in the complete flag variety. There is a certain symmetry between these families, which can be generalized to other submanifolds of the flag variety.