The paper studies polar normalizations of skew ruled surfaces in 3D space.
problem Understanding the properties and invariants of polar normalized skew ruled surfaces.
method Determination of invariants and analysis of Tchebychev and support vector fields.
result Special polar normalizations lead to degenerate curves.
Paper proves extension of unit normal vector field from a hypersurface.
problem Need to extend unit normal vector field from a hypersurface.
method Provides an elementary proof of existence and uniqueness of such an extension.
result Elementary proof of existence and uniqueness of unit gradient field extension.
The normal map of curves is analyzed as a vector field on a cylinder.
problem Understanding the geometric properties of normal maps and their vector field interpretation.
method Interpreting critical points geometrically, studying Poincaré index, projecting to sphere, and analyzing winding and rotation indices.
result Counting theorems regarding winding and rotation indices of curves and their evolutes are proven.
Listed normal forms for metrics with one essential projective vector field.
problem Classifying metrics with a single essential projective vector field.
method Provided a complete list of normal forms for metrics with one essential projective vector field.
result Extended and revised previous classifications by Lie (1882), Bryant & Manno & Matveev (2008), and Matveev (2012).
Study improves self-normalized bounds for vector-valued processes beyond sub-Gaussianity.
problem Limited understanding of self-normalized concentration for vector-valued processes outside sub-Gaussian frameworks.
method Developed concentration inequalities for self-normalized processes with light tails (e.g., Bennett, Bernstein bounds) for vector-valued data.
result Provided new insights and bounds for self-normalized processes with non-sub-Gaussian distributions.
Article studies symmetry in smooth vector bundles using advanced operations.
problem Symmetry phenomena in smooth vector bundles after two iterations of the normal functor.
method Developed theory of pullback and quotient for double vector bundles and morphisms, focusing on naturality of the normal functor.
result Expected symmetry is obtained through universal behavior and compatibility of operations.
New method for mesh denoising using TGV of normal vector field.
problem Improving mesh quality by removing noise.
method Proposes a novel TGV formulation for normal vector fields on triangular meshes.
result New method outperforms existing techniques in mesh denoising experiments.
The article warns against assuming normality in machine learning, especially when feature vectors are not normally distributed.
problem The assumption of normality in machine learning models can be misleading, especially when feature vectors are not normally distributed.
method The article provides a mathematical counterexample and experiments to illustrate the risks of assuming normality.
result Prudence is needed when assuming normality in machine learning models, particularly when feature vectors are not normally distributed.
We consider the Laplace normal vector field of relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the three-dimensional Euclidean space R3. We determine all ruled surfaces and all relative normalizations for which the Laplace normal image degenerates into a point or into a curve…
The paper studies skew ruled surfaces with specific normalization properties.
problem Characterizing skew ruled surfaces with relative normalizations.
method Analyzes skew ruled surfaces in E3 with a specific form of support function. result Investigates properties of Tchebychev and support vector fields for these surfaces.
The paper proposes deep normalization to improve speaker recognition performance.
problem Non-Gaussian and non-homogeneous distributions of deep speaker vectors negatively impact speaker recognition.
method Proposes a deep normalization approach based on a novel discriminative normalization flow (DNF) model.
result DNF-based normalization delivers substantial performance gains and strong generalization capability.
Shows Euler-like vector fields come from specific embeddings.
problem Understanding Euler-like vector fields and their origins.
method Using tubular neighborhood embeddings and normal exponential maps of Riemannian metrics.
result Each Euler-like vector field originates from a specific embedding.
Normal forms and isotropic embeddings via Euler-like vector fields.
problem Proving normal forms results for geometric structures.
method Construction of Euler-like vector fields compatible with geometric structures.
result Illustrated in various examples, including Morse-Bott, Weinstein, and Zung's theorems.
Paper studies CR submanifolds in complex projective space with flat normal connection.
problem Existence and properties of CR submanifolds with flat normal connection.
method Investigation of umbilical normal vector and application to non-existence proof.
result Non-existence of certain CR submanifolds of maximal CR dimension.
Study surfaces with parallel mean curvature in 4D spaces.
problem Characterize surfaces with parallel normalized mean curvature in Euclidean or Minkowski 4-space.
method Introduced special isothermal parameters and described surfaces using invariant functions.
result Surfaces with parallel normalized mean curvature are uniquely determined by three invariant functions.
We prove that a normal vector field along a curve in R3 is rotation minimizing (RM) if and only if it is parallel respect to the normal connection. This allows us to generalize all the results of RM vectors and frames to curves immersed in Riemannian manifolds.
The paper characterizes surfaces in 4D space forms with flat normal connection.
problem Characterizing surfaces in 4D space forms with specific geometric properties.
method Analyzing linearly dependent conditions and using properties of sectional curvature.
result Characterizations of space-like and time-like surfaces with flat normal connection.
The study classifies meridian surfaces with specific curvature properties in a special 4D space.
problem Characterizing surfaces with parallel mean or normalized mean curvature vectors.
method Classification of meridian surfaces based on curvature properties.
result Existence of surfaces with parallel normalized mean curvature but not mean curvature.
In this paper we deal with relative normalizations of hypersurfaces in the (n+1)-dimensional Euclidean space Rn+1. Considering a relative normalization yˉ of an hypersurface Φ we decompose the corresponding Tchebychev vector Tˉ in two components, one parallel to the Tchebychev vector $\bar…
A new fast algorithm for heavy-tailed PLDA improves speaker recognition.
problem Heavy-tailed PLDA is computationally expensive and slower than Gaussian PLDA.
method Introduces a fast, variational Bayes, generative training algorithm for HT-PLDA.
result The new algorithm achieves similar accuracy to HT-PLDA but with reduced computational cost.
The paper generalizes rectifying and normal curves in Lorentzian n-space.
problem Characterizing and classifying g−rectifying and g−normal curves in Lorentzian n-space. method Introducing a g−position vector field and defining g−rectifying and g−normal curves based on this field. result Comprehensive characterization and classification of g−rectifying and g−normal curves. The paper reviews statistical models for high-dimensional normalized vectors.
problem Handling directional data in machine learning.
method Review of mathematical models for normalized vectors on hypersphere and real projective plane.
result Common models and technical aspects are discussed.
We establish normal forms for conformal vector fields on pseudo-Riemannian manifolds in the neighborhood of a singularity. For real-analytic Lorentzian manifolds, we show that the vector field is analytically linearizable or the manifold is conformally flat. In either case, the vector field is locally conjugate to a no…
In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces in the affine space, endowed with a distinguished "relative normal vector field" …
New photometric stereo method using dictionary learning for better normal vector reconstruction.
problem Photometric stereo's reliance on diffuse surface model limits its effectiveness for complex reflectance patterns.
method Developed two formulations of dictionary learning for photometric stereo: one for Lambertian and one for non-Lambertian objects.
result State-of-the-art performance compared to existing robust photometric stereo methods on synthetic and real datasets.
We obtain several rigidity results for biharmonic submanifolds in Sn with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in Sn with parallel normalized mean curvature vector field and with at most two distinct principal curvatures. In particular, we dete…
In this paper we proof that the Holomorphic angle for compact minimal surfaces in the sphere S5 with constant Contact angle and with a parallel normal vector field must be constant.
Study on curves around a Whitney umbrella focusing on geodesic and normal curvatures.
problem Analyzing geometric properties of curves around a specific surface.
method Examined geodesic and normal curvatures, ruled surfaces, and normal developable surfaces.
result Obtained functions representing geometry on a Whitney umbrella.
New Ricci curvature means derived from plane curvatures.
problem Understanding Ricci curvature in geometric contexts.
method Introducing intrinsic and normal mean Ricci curvatures via Jacobi-field expansions and applying Bochner-Weitzenboeck identity.
result Derives a Bochner-Weitzenboeck identity for simple d-vectors.
Minimal totally real submanifolds in complex space forms have special umbilical properties.
problem Characterizing minimal totally real submanifolds in complex space forms.
method Analyzing the position of umbilical normal vectors in the normal bundle.
result Pseudo-umbilical totally real submanifolds with flat normal connection in non-flat complex space forms are minimal.
The study of quotient structures in multi-graded bundles, including double vector bundles.
problem Understanding quotients of multi-graded bundles, especially double vector bundles.
method Analyzing quotients as towers of affine bundles and constructing normal bundles.
result Any quotient of multi-graded bundles fits into a tower of affine bundles.
New equations connect unit Killing vectors to initial data.
problem Characterizing initial data for Einstein vacuum with unit Killing vectors.
method Developed new equations (uKID) by eliminating scaling and using propagation identity.
result Found equations that are finite type and characterize unit normalized Killing vectors.
Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of …
New photometric stereo method using learned dictionaries for robustness.
problem Estimating object normals from varying lighting conditions.
method Adaptive dictionary learning for image preprocessing and direct regularization of normal vectors.
result State-of-the-art performance in noisy conditions.
Physics-based method approximates mean curvature on surface meshes.
problem Estimating mean curvature on triangulated surfaces.
method Derives approximation from Young-Laplace equation and force balance.
result Approximation equivalent to discrete Laplace-Beltrami operator.
I classify spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-euclidean space in arbitrary codimension, if the mean curvature vector is not a null vector and the principal normal vector is parallel in the normal bundle. Moreover, I exclude the existence of such self-shrinkers in several cas…
A new framework enhances generative modeling by learning local flows over complex manifolds.
problem Limited expressivity of current normalizing flows for low-dimensional manifolds.
method Vector quantized local normalizing flows (VQ-Flows) using a VQ-AE atlas and conditional flows.
result Enhanced modeling of complex data distributions over manifolds.
Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.
problem Conservation law for vector fields on surfaces with piecewise smooth boundaries.
method Generalization of the Poincaré-Hopf Theorem for real-analytic vector fields on surfaces with piecewise smooth boundaries.
result Conservation law for vector fields on surfaces with piecewise smooth boundaries.
The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …
Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension …
Classifies singularities of smooth vector fields on the line.
problem Classifying singularities of smooth vector fields on the line.
method Local classification with respect to C1-conjugacy, including normal forms and unfoldings. result Complete description of the 1-d case achieved.
The study characterizes and analyzes spacelike surfaces with a canonical normal null direction in Minkowski 4-space.
problem Characterizing and analyzing spacelike surfaces with a specific null direction in Minkowski space.
method Using geometric properties, Gauss map, and a nonlinear partial differential equation, the study characterizes and analyzes these surfaces.
result Characterizations and properties of spacelike surfaces with a canonical normal null direction are obtained.
The vector of periodic, compound returns of a typical investment portfolio is almost never a convex combination of the return vectors of the securities in the portfolio. As a result the ex post version of Harry Markowitz's "standard mean-variance portfolio selection model" does not apply to compound return data. We pro…
Rectifying curves on hypercones are geodesics, characterized in higher dimensions.
problem Characterizing rectifying curves in higher-dimensional spaces.
method Extending results from Chen (2017) to higher dimensions, using hypercones and hyperplanes.
result Rectifying curves on hypercones are geodesics, and these curves can be mapped to spherical curves in higher dimensions.
Random square-tiled surfaces have normal genus distribution and cover all integer vectors.
problem Distribution and properties of random square-tiled surfaces.
method Randomizing model and local central limit theorem for genus.
result The distribution of the genus is asymptotically normal and contains all primitive integer vectors.
The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The existence of bundle coordinates normal at a given point and/or along injective horizonta…
We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in the case of a …
We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.