New methods find Ricci-flat metrics on specific Lie groups.
problem Finding Ricci-flat metrics on Lie groups.
method Two constructions based on gradings and filtrations.
result Every nilpotent Lie algebra of dimensions up to 9 admits an indefinite Ricci-flat metric.
Study on quasilinear elliptic equation on graphs with indefinite weights.
problem Existence and properties of solutions to quasilinear elliptic equations on graphs with indefinite weights.
method Variational methods and analysis of eigenvalue problems.
result Existence and monotonicity of the principal eigenvalue and positive solutions.
Study on lightlike geometry in indefinite Sasakian statistical manifolds.
problem Exploring lightlike hypersurfaces and their properties in indefinite Sasakian statistical manifolds.
method Introducing indefinite Sasakian statistical manifolds and analyzing lightlike hypersurfaces with respect to dual connections.
result An invariant lightlike submanifold of an indefinite Sasakian statistical manifold is itself an indefinite Sasakian statistical manifold.
In this paper, we show that the calibrated method can also be used to detect indefinite minimal Lagrangian submanifolds in Ckm. We introduce the notion of indefinite special Lagrangian submanifolds in Ckm and generalize the well-known work of Harvey-Lawson to the indefinite case.
Study lightlike submanifolds in indefinite statistical manifolds, finding conditions and curvature expressions.
problem Characterize lightlike submanifolds in indefinite statistical manifolds.
method Analyze conditions for lightlike submanifolds to be lightlike statistical submanifolds, derive statistical sectional curvature, and investigate induced statistical Ricci tensor symmetry.
result Conditions for lightlike submanifolds to be lightlike statistical submanifolds and expressions for statistical sectional curvature and induced Ricci tensor symmetry.
A Morse 2-function is a generic smooth map from a smooth manifold to a surface. In the absence of definite folds (in which case we say that the Morse 2-function is indefinite), these are natural generalizations of broken (Lefschetz) fibrations. We prove existence and uniqueness results for indefinite Morse 2-functions …
We present a compared analysis of some properties of indefinite almost S-manifolds and indefinite S-manifolds. We give some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields. We study the sectional and φ-sectional curvature of indefinite almost $\…
Paper converts deep networks to flat, equivalent kernel machines.
problem Capacity control and uniform convergence in deep learning.
method Push-forward transformation from deep networks to indefinite kernel machines.
result Flat network weights are Lp-norm regularized (0<p<1).
Study shows lightlike hypersurfaces in indefinite Sasakian manifolds are not symmetric.
problem Characterizing curvature symmetries in lightlike hypersurfaces of indefinite Sasakian manifolds.
method Analyzing properties of lightlike hypersurfaces in indefinite Sasakian manifolds.
result Lightlike hypersurfaces are not locally symmetric, semi-symmetric, or semi-parallel.
An η-Einstein condition is introduced in the context of indefinite g.f.f-manifolds, and a few Schur-type lemmas for indefinite S-manifolds are provided.
Study on special null submanifolds in indefinite Sasakian manifolds.
problem Characterizing null submanifolds in indefinite Sasakian manifolds.
method Proving properties of screen conformal null submanifolds and defining a new class.
result Existence of contact screen conformal r-null submanifolds in indefinite Sasakian space forms. Formula derived for discrete improper affine spheres.
problem Constructing discrete improper affine spheres.
method Loop group factorizations and Birkhoff decomposition.
result Representation formula for discrete indefinite affine spheres.
This paper extends stability analysis to non-convergent neural network training.
problem Generalization of neural networks whose training does not converge to fixed points.
method Introduces statistical algorithmic stability (SAS) to study non-convergent algorithms and their generalization.
result Stability of non-convergent training dynamics correlates with generalization performance.
Study connects spectral geometry with Coulomb interactions in perforated manifolds.
problem Understanding spectral properties of perforated manifolds and their interactions.
method Optimal convergence rates for Steklov eigenvalues and expansions, derived from Green function and Coulomb-type energy.
result Identified two correction scales for Steklov eigenvalues in dimensions two and three.
New Einstein metric found on non-standard solvmanifold.
problem Finding indefinite invariant Einstein metrics on solvmanifolds.
method Described an example of a non-standard solvmanifold with an indefinite invariant Einstein metric.
result Found a new type of Einstein metric on a non-standard solvmanifold.
We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reve…
Study proves no lightlike hypersurfaces exist in certain indefinite Sasakian manifolds.
problem Existence of lightlike hypersurfaces in indefinite Sasakian manifolds.
method Proved non-existence through properties of second fundamental forms and induced structural tensors.
result No lightlike hypersurfaces can have parallel or recurrent second fundamental forms or induced structural tensors.
Study on SASI-lightlike submanifolds in indefinite Kaehler manifolds.
problem Characterizing and understanding SASI-lightlike submanifolds in indefinite Kaehler manifolds.
method Introduced and analyzed SASI-lightlike submanifolds, derived conditions for metric connection, and investigated integrability of distributions.
result Found necessary and sufficient conditions for the induced connection to be a metric connection on SASI-lightlike submanifolds.
The paper studies singularities in discrete indefinite affine minimal surfaces.
problem Characterizing singularities in discrete indefinite affine minimal surfaces.
method Discretizing smooth curves and applying discrete Lelieuvre's formulas to study the resulting surfaces.
result The definition of singular edges and vertices in discrete asymptotic nets mirrors properties of smooth surfaces.
Introduces a variational framework for indefinite Lagrangians with specific symmetries.
problem Handling indefinite Lagrangians with complex symmetries.
method Develops a variational setting for an indefinite Lagrangian with a specific Noether charge.
result Validates the existence of a variational setting for a broad class of Lagrangians.
We study the basic geometric properties of an indefinite locally conformal Kaehler manifold.
New indefinite false theta functions match homological blocks for a specific 3-manifold.
problem Constructing homological blocks for certain 3-manifolds.
method Developing indefinite false theta functions and proving their equivalence to homological blocks.
result Indefinite false theta functions match homological blocks for the Poincaré homology sphere.
We improve our previous results on indefinite Kasparov modules, which provide a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. In particular, we can weaken the assumptions that are imposed on indefinite Kasparov modules. Using a new theorem by Lesch an…
We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…
WIPS optimizes inner product weights to approximate various similarities.
problem Learning high-quality node representations and accurate similarities.
method Weighted inner product similarity (WIPS) with adjustable weights.
result WIPS can approximate arbitrary general similarities including positive definite and indefinite kernels.
In this paper we build the structure equations and the integrable systems for a discrete centroaffine indefinite surface in R3. At the same time, some centroaffine invariants are obtained according to the structure equations. Using these centroaffine invariants, we study the Laplacian operator and the convexity of …
We construct a new representation formula for indefinite improper affine spheres in terms of two para-holomorphic functions and study singularities which appear in this representation formula. As a result, it follows that cuspidal cross caps never appear as the singularities on indefinite improper affine spheres and so…
The study explores indefinite nilsolitons and Einstein solvmanifolds, revealing new geometric properties.
problem Understanding indefinite nilsolitons and their relation to Einstein solvmanifolds.
method Analyzing algebraic properties and geometric structures of nilpotent Lie algebras.
result Established a correspondence between indefinite Einstein solvmanifolds and nilsolitons, differing from the Riemannian case.
Enhances KLR for indefinite kernels with L1-norm regularization.
problem Classifying with indefinite kernels captures more domain-specific information.
method Introduces L1-norm regularization to induce sparsity and a proximal linearized algorithm. result Superior performance in accuracy and sparsity on multiple datasets.
Indefinite Kaehler solutions of the Einstein equations are studied, and it is almost completely determined which compact complex surfaces admit such metrics.
Study on null submanifolds in indefinite complex contact geometry.
problem Geometry of null submanifolds in indefinite complex contact manifolds.
method Analysis of quaternion null submanifolds and distributions on screen submanifolds.
result Quaternion null submanifolds are always totally geodesic.
In a metric g.f.f-manifold we study lightlike hypersurfaces M tangent to the characteristic vector fields, and owing to the presence of the f-structure, we determine some decompositions of TM and of a chosen screen distribution obtaining two distributions invariant with respect to the structure. We discuss the …
Some curvature properties of Kahler manifolds of indefinite metrics are studied. Analogues of a Kulkarni's theorem are proved for such manifolds.
We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we classify the helicoidal indefinite improper affine spheres and find a new family…
Study of spheres and circles on a manifold with a specific metric structure.
problem Understanding geometric objects on a manifold with a skew-circulant structure.
method Analyzing hyper-spheres, spheres, and circles in a tangent space of a 4D manifold with a skew-circulant tensor structure.
result Characterization of geometric objects under an indefinite metric.
Since J. L. Lagrange initiated in 1760 the study of minimal surfaces of Euclidean 3-space, minimal surfaces in real space forms have been studied extensively by many mathematicians during the last two and half centuries. In contrast, so far very few results on minimal Lorentz surfaces in indefinite space forms are know…
New method reduces variance and bias in approximating indefinite kernels.
problem Approximating non-stationary indefinite kernels with low variance and bias.
method Generalized orthogonal random features (GORF)
result GORF achieves lower variance and approximation error compared to existing methods.
Study on lightlike submanifolds in statistical manifold geometry.
problem Characterizing contact CR and SCR-lightlike submanifolds.
method Developed characterization theorems on integrability and geodesicity.
result Obtained results on geometry of contact CR and SCR-lightlike submanifolds.
The paper bounds the genus of surfaces in four-manifolds with indefinite forms.
problem Bounding the genus of surfaces in four-manifolds with indefinite intersection forms.
method Using adjunction inequalities and the 10/8 + 4 theorem, the paper provides methods to bound the genus of surfaces.
result The set of H-slice knots can detect exotic smooth structures on closed 4-manifolds.
We show that an indefinite Euclidean complex space is not a relative of an indefinite non-flat complex space form. We further study whether two compact Fubini-Study spaces are relatives or not.
Study geometric properties of SGL submanifolds in a specific manifold.
problem Analyzing geometric characteristics of SGL submanifolds.
method Examines integrability conditions and parallelism properties of distributions.
result Provides insights into geometric behavior of SGL submanifolds.
Twistor correspondences for R-invariant indefinite self-dual conformal structures on R^4 are established explicitly. These correspondences are written down by using a natural integral transform from functions on a two dimensional cylinder to functions on the flat Lorentz space R^{1,2} which is related to the wave equat…
Minimal number of geodesics in Finsler manifolds with indefinite Killing form is at least four.
problem Determining the minimal number of homogeneous geodesics in Finsler manifolds with indefinite Killing form.
method Analyzing examples of Lie groups with invariant Finsler metrics and presenting new examples.
result Homogeneous Finsler manifolds with indefinite Killing form admit at least four homogeneous geodesics.
The study extends Jacobi-orthogonality to indefinite scalar product spaces.
problem Generalizing Jacobi-orthogonality to indefinite scalar product spaces.
method Comparing principles, investigating tensor relations, proving properties.
result Every quasi-Clifford tensor is Jacobi-orthogonal; certain tensors are Jacobi-dual or Osserman.
The paper studies timelike minimal Lagrangian surfaces in indefinite complex hyperbolic space.
problem Characterizing timelike minimal Lagrangian surfaces in indefinite complex hyperbolic space.
method Defining natural Gauss maps and proving a Ruh-Vilms type theorem.
result Timelike minimal Lagrangian surfaces correspond to the fifth real form of the complex affine Kac-Moody algebra.
The study proves that pseudo-umbilical spacelike submanifolds are totally geodesic and totally umbilical.
problem Characterizing properties of pseudo-umbilical spacelike submanifolds in indefinite space forms.
method Derived an intrinsic inequality and used it to show total geodesic property. Proved total umbilicity using Aiyama's result.
result Pseudo-umbilical spacelike submanifolds are totally geodesic and totally umbilical.
This paper studies ruled real hypersurfaces in indefinite complex projective space.
problem Characterizing and classifying ruled real hypersurfaces in indefinite complex projective space.
method Introduced and studied ruled real hypersurfaces with maximal holomorphic distribution integrable and leaves totally geodesic holomorphic hyperplanes. Detailed shape operator computation and method of construction by gluing totally geodesic hyperplanes along a curve.
result Classification of all minimal ruled real hypersurfaces in terms of three main families of curves.
It is well known that if the dimension of the Sasaki cone is greater than one, then all Sasakian structures are either positive or indefinite. We discuss the phenomenon of type changing within a fixed Sasaki cone. Assuming henceforth that the dimension of the Sasaki cone is greater than one, there are three possibiliti…