Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.
problem Characterizing homogeneous Lorentzian three-manifolds with a 4D isometry group.
method Explicit global coordinate description and proof of Ricci soliton properties.
result All special examples are non-gradient expanding Ricci solitons.
Study on non-gradient Ricci almost solitons in warped products.
problem Understanding non-gradient Ricci almost solitons.
method Construction method and explicit example in warped products.
result Rigidity result for Gaussian soliton.
New degree theory proves existence of solitons on 4D manifolds.
problem Existence of gradient expanding solitons on 4D manifolds.
method Developed new degree theory for 4D, asymptotically conical gradient expanding solitons.
result Existence of solitons asymptotic to any cone over S^3 with non-negative scalar curvature.
The paper classifies Ricci solitons and studies harmonic vector fields on a specific Thurston geometry.
problem Classifying Ricci solitons and studying harmonic vector fields in a specific Thurston geometry.
method Left-invariant Riemannian metric classification and analysis of harmonic maps and vector fields.
result All Ricci solitons on (F4,g) are expanding and non-gradient. Study of Bach flow on specific nilmanifolds, converging to a soliton.
problem Analyzing the Bach flow on specific nilmanifolds.
method Fourth order geometric flow on four-dimensional simply connected nilmanifolds.
result The Bach flow converges to an expanding Bach soliton on these manifolds.
With a f-left-invariant Riemannian metric on a Lie group G, we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor f. In this article, we study the geometry of such metrics and give a necessary and sufficient condition for an f-left-invariant Rie…
Gradient Ricci solitons can be extended to non-gradient Ricci solitons using energy function.
problem Extending the geometry of gradient Ricci solitons to non-gradient Ricci solitons.
method Using energy function E to study the geometry. result A non-steady Ricci soliton with symmetric covariant derivative is gradient.
The paper classifies quasi-Einstein 3-manifolds and their properties.
problem Classifying compact locally homogeneous non-gradient quasi-Einstein 3-manifolds.
method Analyzing quotient spaces of Lie groups and using properties of quasi-Einstein metrics.
result Identifies conditions for the existence of nontrivial quasi-Einstein metrics.
A1GM method improves efficiency in reconstructing missing data using KL divergence.
problem Efficiently reconstructing missing data in matrices.
method Fast non-gradient-based rank-1 NMF using KL divergence.
result A1GM outperforms gradient methods in efficiency with competitive reconstruction errors.
PairNets optimize AI models for fast IoT applications.
problem Slow training and high memory usage of deep neural networks.
method Developed Pairwise Neural Networks (PairNets) with low memory and fast training.
result PairNets achieve faster training (one epoch) and lower prediction errors.
In this work, a non-gradient descent learning (NGDL) scheme was proposed for deep feedforward neural networks (DNN). It is known that an autoencoder can be used as the building blocks of the multi-layer perceptron (MLP) DNN, the MLP is taken as an example to illustrate the proposed scheme of pseudoinverse learning algo…
This research analyzes how input and output layers affect deep neural networks' resistance to adversarial attacks.
problem The vulnerability of deep neural networks to adversarial inputs, especially non-gradient based attacks.
method Analysis of three different fully connected dense network classes with manipulated input and output layers.
result Manipulating input and output layers can significantly enhance a deep neural network's robustness against adversarial attacks.
Study defends shallow neural networks from data-poisoning attacks.
problem Protecting shallow neural networks from adversarial attacks during training.
method Developed a non-gradient stochastic algorithm for depth-2 neural networks, proving near-optimal trade-offs.
result Demonstrated improved performance over stochastic gradient descent under various data distributions.
We study 3-dimensional Ricci solitons which project via a semi-conformal mapping to a surface. We reformulate the equations in terms of parameters of the map; this enables us to give an ansatz for constructing solitons in terms of data on the surface. A complete description of the soliton structures on all the 3-di…
The three-dimensional Heisenberg group H3 has three left-invariant Lorentz metrics g1, g2 and g3. They are not isometric each other. In this paper, we characterize the left-invariant Lorentzian metric g1 as a Lorentz Ricci soliton. This Ricci soliton g1 is a shrinking non-gradient Ricci soliton. Likew…
Paper studies non-gradient almost Yamabe solitons and their properties.
problem Characterizing structures of non-gradient almost Yamabe solitons.
method Investigates conditions for trivial solitons and local warped product structures.
result Almost Yamabe solitons with closed vector fields admit local warped product structures.
In this paper, we briefly review the basic scheme of the pseudoinverse learning (PIL) algorithm and present some discussions on the PIL, as well as its variants. The PIL algorithm, first presented in 1995, is a non-gradient descent and non-iterative learning algorithm for multi-layer neural networks and has several adv…
A general Boltzmann machine with continuous visible and discrete integer valued hidden states is introduced. Under mild assumptions about the connection matrices, the probability density function of the visible units can be solved for analytically, yielding a novel parametric density function involving a ratio of Riema…
New quasi-Einstein metrics found on a sphere.
problem Finding quasi-Einstein metrics on a sphere.
method Constructing axi-symmetric non-gradient m-quasi-Einstein structures using hypergeometric functions. result Found new regular metrics on a two-sphere, including the extreme Kerr black hole horizon.
Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
New expanders found using origami surfaces with spectral gap.
problem Constructing expanders with spectral gap on surfaces of arbitrary genus.
method Affine actions on origami surfaces to achieve spectral gap.
result New expanders distinct from classical ones.
The paper extends rigidity results for λ-self-expanders to hyperplanes, spheres, and cylinders.
problem Characterizing λ-self-expanders as hyperplanes, spheres, and cylinders. method Extending results on self-expanders to λ-self-expanders, proving rigidity results. result Characterizes hyperplanes, spheres, and cylinders as λ-self-expanders. New self-expander found between two given asymptotic ones.
problem Finding new self-expanders between given asymptotic ones.
method Developed a min-max theory for asymptotically conical self-expanders of mean curvature flow.
result Existence of a new asymptotically conical self-expander trapped between two given ones.
No expanding breathers found in noncompact Ricci flows with certain curvature conditions.
problem Finding expanding breathers in noncompact Ricci flows.
method Curvature positivity conditions (weak PIC-2 or nonnegative bisectional curvature).
result Complete noncompact expanding breathers are gradient solitons.
SGLD proves geometric ergodicity via reflection coupling for nonconvex log-concave distributions.
problem Proving geometric ergodicity of SGLD in nonconvex, log-concave settings.
method Reflection coupling technique to handle SGLD's time discretization and minibatch issues.
result SGLD has an invariant distribution and geometric ergodicity in W1 distance. New expanders for mean curvature flow contradict genus-reduction conjecture.
problem Contradicting Ilmanen's genus-reduction conjecture for mean curvature flow.
method Construct new expanders asymptotic to cones arising from shrinkers.
result Existence of expanders of arbitrarily large genus.
New expanding Ricci solitons found starting in dimension four.
problem Finding expanding Ricci solitons in specific dimensions.
method Constructing gradient expanding Ricci solitons asymptotic to cones and on trivial vector bundles.
result Continuous families of expanding Ricci solitons on products of Einstein manifolds.
Study on the spectrum of drift Laplacian on Ricci expanders.
problem Analyzing the spectrum of the drift Laplacian on Ricci expanders.
method Investigation of discrete spectrum under proper potential function, asymptotic behavior of potential function, and computation of eigenvalues.
result Discrete spectrum of the drift Laplacian on Ricci expanders with bounded Ricci curvature.
Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expand…
The paper classifies expanding gradient Yamabe solitons based on scalar curvature.
problem Classifying expanding gradient Yamabe solitons based on scalar curvature.
method Rigorous analysis of scalar curvature in both cases: greater than and less than the soliton constant.
result Complete classification of nontrivial complete expanding gradient Yamabe solitons.
Constructs self-expanders of positive genus for cones in R^3.
problem Creating self-expanders of positive genus for cones in R^3.
method Constructs self-expanders asymptotic to cones, uses mean curvature flow.
result Constructs self-expanders with unbounded genus asymptotic to a rotationally symmetric cone.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
Study finds unique self-expanders for mean curvature flow.
problem Finding solutions to mean curvature flow.
method Derived equation based on generalized Lawson-Osserman cone and modified equilibria theory.
result Existence and uniqueness of self-expanders proved.
We derive a sharp lower bound for the scalar curvature of non-flat and non-compact expanding gradient Ricci soliton provided that the scalar curvature is non-negative and the potential function is proper. We also give an upper bound for the scalar curvature of noncompact expander when the Ricci curvature is nonpositive…
Study cohomogeneity one expanding Ricci solitons on specific topologies.
problem Characterize and analyze cohomogeneity one expanding Ricci solitons.
method Analyze ODEs, define expander degree, calculate cohomogeneity one expander degree.
result Reconstruct and calculate cohomogeneity one expander degree for specific topologies.
We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A.
We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. …
The paper examines properties and rigidity of self-expanders in Euclidean space.
problem Characterizing and estimating properties of self-expanders in Euclidean space.
method Analyzing mean curvature flow, volume growths, and stability of self-expanders.
result Proves the uniqueness of certain self-expanders in 3D space.
The paper classifies 2D complete Lagrangian self-expanders in complex 2-space.
problem Classifying complete Lagrangian self-expanders in complex 2-space.
method Obtained a classification theorem.
result A classification of 2D complete Lagrangian self-expanders with constant squared norm of the second fundamental form.
Strict convexity proven for certain self-expanders in high dimensions.
problem Convexity of self-expanders in mean curvature flow.
method Investigation of convexity properties for asymptotically conical self-expanders.
result Strict convexity proven for n-dimensional self-expanders. We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional self-expanders of a fixed topological type and, in all dimensions, for the set of self-expa…
Study classifies 4D Ricci solitons with specific curvature conditions.
problem Classifying 4D gradient steady and expanding Ricci solitons with given curvature properties.
method Asymptotically cylindrical and conical assumptions; half-harmonic and half-nonnegative isotropic curvature conditions.
result Partial classification of 4D gradient expanding Ricci solitons with half-nonnegative isotropic curvature.
We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that non-degenerate self-expanders -- that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone -- are generic in a certain …
We define a way of approximating actions on measure spaces using finite graphs; we then show that in quite general settings these graphs form a family of expanders if and only if the action is expanding in measure. This provides a somewhat unified approach to construct expanders. We also show that the graphs we obtain …
In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to O(n)-invariant co…
Proves uniqueness of small entropy self-expanders.
problem Uniqueness of self-expanders with small entropy.
method Mountain-pass theorem and integer degree argument.
result Proves uniqueness result for self-expanders.
Study self-expanding solutions of mean curvature flow in various dimensions.
problem Characterize complete mean convex self-expanding hypersurfaces and their properties.
method Analyzing the function ∣A∣2/∣H∣2 and ∣Aξ∣2/∣H∣2 to understand the structure of self-expanders. result Complete mean convex self-expanders are products of self-expanding curves and flat subspaces under certain conditions.
We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that expanders with vanishing relative entropy are unique in a generic sense. This also impl…