Special Manifolds Research Articles

Manifold-based approach for neural network robustness analysis

It is important to understand the mathematical foundations of neural networks and to include robustness in model evaluation. Here, we introduce algorithms based on manifold curvature estimation to assess neural network robustness. These algorithms rely solely on training data and do not require regular or adversarial test data. Initially, a metric is proposed to measure the curvature of discrete data manifolds by introducing weighted angles concept between subspaces. Following this, a robustness measure is introduced that is independent of network architecture or model parameters. Lastly, two additional methods are introduced, utilizing curvature estimation of special manifolds formed by using gradient vectors between output and input network layers, alongside manifold curvature estimation. A comprehensive evaluation is provided on multiple network models using the CIFAR-10 dataset. Manifold geometry-based robustness analysis may lead to the development of not only accurate but also robust neural network models.

Multi-label deep transfer learning method for coupling fault diagnosis

With the development of complexity and integration of machines, the multiple components of the system are prone to simultaneous failures and the multi-fault signals may be measured in limited sensors. This fault mode is defined as coupling fault in this paper and becomes more common and attracts great attention. Current coupling fault diagnosis approaches rely on labeled data under consistent working conditions. However, in practical varying working conditions, the data is difficult to label and the traditional methods perform poorly. To address the challenge problem, we utilize some related annotated data under other working conditions for auxiliary learning and abstract this problem as a multi-label transfer learning problem which is a new research topic in fault diagnosis. We proposed a novel deep transfer coupling fault diagnosis method from the global and local views to achieve the effect of transfer, which is the first attempt to address this practical problem as far as we know. At the global level, we used Maximum Mean Discrepancy (MMD) at multiple stages of the network to reduce the hypothesis space of the model. Furthermore, we design a special manifold framework with multi-level similarity to constrain the hierarchy of the distance of the sample and the hierarchy of the labels being consistent between the two domains, to further improve the effect of local alignment. The proposed method has high accuracy in multiple transfer tasks for both public and laboratory datasets, powerfully demonstrating its effectiveness.

Albanese map of special manifolds: a correction

Albanese map of special manifolds: a correction

Similarity revisited: shock random walks in the asymmetric simple exclusion process with open boundaries

A reverse duality between the asymmetric simple exclusion process (ASEP) with open boundary conditions and a biased random walk of a single particle is proved for a special manifold of boundary parameters of the ASEP. The duality function is given by the configuration probabilities of a family of Bernoulli shock measures with a microscopic shock at site x of the lattice. The boundary conditions of the dual random walk, which can be reflecting or absorbing, depend on the choice of the duality function. As a consequence of this duality, the full time-dependent distribution of the open ASEP starting from a fixed shock measure at x is given for any time t>0 by a convex combination of shock measures with shock at position y. The coefficients are the transition probabilities P(x, t|y, 0) at time t of the random walk starting at site y. It is also shown that this family of shock measures arises from a similarity transformation of the two-dimensional representation of the matrix algebra for the stationary matrix product measure of the open ASEP. This implies a further duality between random walks with different boundary conditions.

Nontrivial solutions of modified nonlinear fourth-order elliptic equation in [formula omitted

This paper is concerned with the following fourth-order elliptic equation of the form: Δ2u−Δu−uΔ(u2)+u=|u|p−2u,x∈RN,where Δ2u≔Δ(Δu), 4≤N≤6 and p∈2,2∗. By making use of the constrained minimization on a special manifold, we prove that the existence of nontrivial solutions of the above problem for any p∈2,2∗. Our result can be regarded as a partial extension of some related literature with 4<p<2∗.

Design and Thermal Analysis of a 3-D Printed Impingement Pin Fin Cold Plate for Heterogeneous Integration Application

Miniaturization high heat flux of power electronic devices have posed a colossal challenge for adequate thermal management. Conventional air-cooling solutions are inadequate for high-performance electronics. Liquid cooling is an alternative solution due to the higher specific heat and latent heat associated with the coolants. Liquid-cooled cold plates are typically manufactured by different approaches, such as skived, forged, extrusion, and electrical discharge machining. When researchers are facing challenges in creating complex geometries in small spaces, 3-D-printing can be a solution. In this article, a 3-D-printed cold plate was designed and characterized with water coolant. The printed metal fin structures were strong enough to undergo pressure from the fluid flow even at high flow rates and small fin structures. A copper block with the top surface area of 1 in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times \,\,1$ </tex-math></inline-formula> in was used to mimic a computer chip. Experimental data have good match with a simulation model, which was built using commercial software 6SigmaET. Effects of geometry parameters and operating parameters were investigated. The fin diameter was varied from 0.3 to 0.5 mm and the fin height was maintained at 2 mm. A special manifold was designed to maximize the surface contact area between coolant and metal surface and therefore minimize thermal resistance. The flow rate was varied from 0.75 to 2 L/min and the coolant inlet temperature was varied from 25 °C to 48 °C. It was observed that for the coolant inlet temperature 25 °C and aluminum cold plate, the junction temperature was kept below 63.2 °C at an input power of 350 W and the pressure drop did not exceed 23 Kpa. Effects of metal materials used in 3-D-printing on the thermal performance of the cold plate were also studied in detail.

Generic anisotropic Lifshitz scalar field theory: New massive minimal subtraction

We formulate the simplest minimal subtraction version for massive scalar fields with O(N) symmetry for generic anisotropic Lifshitz spacetimes. We introduce the simplest geometric concepts to define it as a special manifold. Then we restrict ourselves to flat Euclidean spaces. An appropriate partial-p operation is applied in the bare two-point vertex function diagrams, which separates the original diagram into a sum of two different integrals which are the coefficients of the corresponding polynomials in the mass and external momentum. Within the proposed method, the coefficient of the mass terms can be discarded and we obtain a minimal subtraction method almost identical to the same scheme in the massless theory in every external momentum/mass subspace. We restrict our demonstration of the method up to three-loop order in the two-point vertex part. We verify its consistency through a diagrammatic computation of static critical exponents, which validates the universality hypothesis.

Positive radial solutions for a class of quasilinear Schrödinger equations in R 3

This paper is concerned with the following quasilinear Schrödinger equations of the form: − Δ u − u Δ ( u 2 ) + u = | u | p − 2 u , x ∈ R 3 , where p ∈ ( 2 , 12 ) . By making use of the constrained minimization method on a special manifold, we prove that the existence of positive radial solutions of the above problem for any p ∈ ( 2 , 12 ) .

Automatic detection of epileptic seizures using Riemannian geometry from scalp EEG recordings.

This paper proposes a new framework for epileptic seizure detection using non-invasive scalp electroencephalogram (sEEG) signals. The major innovation of the current study is using the Riemannian geometry for transforming the covariance matrices estimated from the EEG channels into a feature vector. The spatial covariance matrices are considered as features in order to extract the spatial information of the sEEG signals without applying any spatial filtering. Since these matrices are symmetric and positive definite (SPD), they belong to a special manifold called the Riemannian manifold. Furthermore, a kernel based on Riemannian geometry is proposed. This kernel maps the SPD matrices onto the Riemannian tangent space. The SPD matrices, obtained from all channels of the segmented sEEG signals, have high dimensions and extra information. For these reasons, the sequential forward feature selection method is applied to select the best features and reduce the computational burden in the classification step. The selected features are fed into a support vector machine (SVM) with an RBF kernel to classify the feature vectors into seizure and non-seizure classes. The performance of the proposed method is evaluated using two long-term scalp EEG (CHB-MIT benchmark and private) databases. Experimental results on all 23 subjects of the CHB-MIT database reveal an accuracy of 99.87%, a sensitivity of 99.91%, and a specificity of 99.82%. In addition, the introduced algorithm is tested on the private sEEG signals recorded from 20 patients, having 1380 seizures. The proposed approach achieves an accuracy, a sensitivity, and a specificity of 98.14%, 98.16%, and 98.12%, respectively. The experimental results on both sEEG databases demonstrate the effectiveness of the proposed method for automated epileptic seizure detection, especially for the private database which has noisier signals in comparison to the CHB-MIT database. Graphical Abstract Block diagram of the proposed epileptic seizure detection algorithm.

Projective Curvature Tensors of Some Special Manifolds with Non-symmetric Linear Connection

We consider geodesic mappings between some special manifolds with the non-symmetric linear connection. We obtain some tensors that are invariant with respect to geodesic mappings by relaxing the so-called “equitorsion condition”. Derived tensors have algebraic expressions analogous to the Weyl tensor of projective curvature, apart from that the generalized Ricci and curvature tensors appeared instead of the usual Ricci and curvature tensors. Further, we study some properties of mentioned invariant tensors as well as their relations with the Weyl tensor of projective curvature. Finally, we introduce some kinds of recurrent manifolds with the non-symmetric linear connection as well as their generalized Ricci and projective counterparts.

On the existence of ground states of an equation of Schrödinger–Poisson–Slater type

We study the existence of ground states of a Schrödinger–Poisson–Slater type equation with pure power nonlinearity. By carrying out the constrained minimization on a special manifold, which is a combination of the Pohozaev manifold and Nehari manifold, we obtain the existence of ground state solutions of this system.

Geodesics of a Special Multiply Warped Product Manifolds

Special multiply warped product manifold as the generalization of a singly warped product manifold which plays an important role in the geometry and physics were considered. Essentially geodesics of this multiply warped product manifold have been presented.

On the Mean Curvature Flow of Submanifolds in the Standard Gaussian Space

In this paper, we study the regular geometric behavior of the mean curvature flow (MCF) of submanifolds in the standard Gaussian metric space $$({{\mathbb {R}}}^{m+p},e^{-|x|^2/m}\overline{g})$$ where $$({{\mathbb {R}}}^{m+p},\overline{g})$$ is the standard Euclidean space and $$x\in {{\mathbb {R}}}^{m+p}$$ denotes the position vector. Note that, as a special Riemannian manifold, $$({{\mathbb {R}}}^{m+p},e^{-|x|^2/m}\overline{g})$$ has an unbounded curvature. Up to a family of diffeomorphisms on $$M^m$$ , the mean curvature flow we considered here turns out to be equivalent to a special variation of the “conformal mean curvature flow” which we have introduced previously. The main theorem of this paper indicates, geometrically, that any immersed compact submanifold in the standard Gaussian space, with the square norm of the position vector being not equal to m, will blow up at a finite time under the mean curvature flow, in the sense that either the position or the curvature blows up to infinity; Moreover, by this main theorem, the interval [0, T) of time in which the flowing submanifolds keep regular has some certain optimal upper bound, and it can reach the bound if and only if the initial submanifold either shrinks to the origin or expands uniformly to infinity under the flow. Besides the main theorem, we also obtain some other interesting conclusions which not only play their key roles in proving the main theorem but also characterize in part the geometric behavior of the flow, being of independent significance.

Multi-geometric Sparse Subspace Clustering

Recently, the Riemannian manifold has received special attention in unsupervised clustering since the real-world visual data usually resides on a special manifold where Euclidean geometry fails to capture. Although many clustering algorithms have been proposed, most of them use only a single geometric model to describe the data. In this paper, a multi-geometric subspace clustering model is proposed, and the subspace representation is learned together by constructing a shared affinity matrix of multi-order data. Experimental results on several different types of datasets show that the clustering performance of our proposed algorithm is better than most of subspaces algorithms.

Special holonomy manifolds, domain walls, intersecting branes and T-folds

We discuss the special holonomy metrics of Gibbons, Lu, Pope and Stelle, which were constructed as nilmanifold bundles over a line by uplifting supersymmetric domain wall solutions of supergravity to 11 dimensions. We show that these are dual to intersecting brane solutions, and considering these leads us to a more general class of special holonomy metrics. Further dualities relate these to non-geometric backgrounds involving intersections of branes and exotic branes. We discuss the possibility of resolving these spaces to give smooth special holonomy manifolds.

Positive solutions for nonlinear Schrödinger–Kirchhoff equations in [formula omitted

In this paper, we study the nonlinear Schrödinger–Kirchhoff-type equation with pure power nonlinearities in R3 by variational methods. By carrying out the constrained minimization on a special manifold which is a combination of the Nehari manifold and Pohozaev manifold, we proved the existence of positive radial solutions of this equation for the power p∈(1,5). The results of this paper extend some existing conclusions, especially for p∈(1,2].

Cross-covariance matrix: Time-shifted correlations for 3D action recognition

In this paper, we introduce cross-covariance to form Symmetric Positive Definite (SPD) matrix-based representations for 3D action recognition. The cross-covariance is generated by the correlational statistics between the time-shifted poses, which brings more informative features and time-order structure to improve the discriminative power on actions. Due to the special manifold structure, the cross-covariance is a totally different SPD matrix representation compared to covariance. In order to utilize such cross-covariance statistics while holding the covariance statistics, we devise a fashion of expression to mix them together as a larger SPD matrix benefiting from the Riemannian geometry. Further, we extend the symmetric cross-covariance into the non-symmetric versions, where the time-order information can be embodied in the related matrix structures. Thus, the derived SPD representations are able to improve the discriminative power of actions. In addition, our representations are easy to implement and to be extended by other prevalent techniques, such as kernel methods, metric learning, etc. Experimental results show that our proposed method efficiently promotes the performance of 3D action recognition tasks. In some public databases, our method outperforms the state-of-the-art methods which are based on other feature representations, kernel matrices, temporal hierarchical features, etc.

Обобщенные многообразия Кенмоцу постоянного типа

В работе мы рассматриваем обобщенные многообразия Кенмоцу, мы вводим четвертое и пятое фундаментальные тождества обобщенных многообразий Кенмоцу, вводятся первый и второй структурные тензоры обобщенных многообразий Кенмоцу и доказаны их свойства, вводится понятие присоединенной Q-алгебры для обобщенных многообразий Кенмоцу. Доказано, что обобщенное многообразие Кенмоцу, а также специальные обобщенные многообразия Кенмоцу II рода имеют антикоммутативную присоединенную Q-алгебру.А многообразия Кенмоцу и специальные обобщенные многообразия Кенмоцу I рода имеют абелеву присоединенную Q-алгебру. Вводится контактный аналог постоянства типа и подробно исследуются обобщенные многообразия Кенмоцу постоянного типа. Получены условия точечного постоянства типа обобщенных многообразий Кенмоцу на пространстве присоединенной G-структуры. Доказано, что класс GK-многообразий нулевого постоянного типа совпадает с классом многообразий Кенмоцу, а класс GK-многообразий ненулевого постоянного типа конциркулярным преобразованием переводит-ся в почти контактное метрическое многообразие локально эквивалентное произведению шестимерного собственногоNK-многообразия на вещественную прямую.

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.

Rational approximation to surfaces defined by polynomials in one variable

We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can be viewed as a Cartesian product of polynomial curves and it is possible to generalize recent results concerning such curves with similar concepts. There is hope that the method leads to insights on how to treat more general manifolds defined by arbitrary polynomials with rational coefficients.