Study defines new slant ruled surfaces in Minkowski 3-space.
problem Characterizing non-null ruled surfaces in Minkowski 3-space.
method Introduced new types of non-null ruled surfaces and their characterizations.
result Established relationships between non-null slant ruled surfaces and their striction lines.
Defines new ruled surfaces using alternative frames and analyzes their geometric properties.
problem Analyzing geometric properties of new ruled surfaces.
method Using alternative frames to define new ruled surfaces and studying their differential geometric properties.
result Geodesic curvatures, normal curvatures, and geodesic torsions of the base curve and striction line are found.
In this study we give definitions and characterizations of transversal surfaces of timelike ruled surfaces. We study some special cases such as the striction curve is a geodesic, an asymptotic line or a line of curvature. Moreover, we obtain developable conditions for transversal surfaces of a timelike ruled surface.
In this study, we investigate the existence theorems for timelike ruled surfaces in Minkowski 3-space. We obtain a general system and give the existence theorems for a timelike ruled surface according to Gaussian curvature, distribution parameter and strictional distance. Moreover, we give some special cases such as th…
In this paper, we consider non developable ruled surface with spacelike ruling, timelike ruling, respectively. We give the relations between the structure functions with the curvature and torsion of the striction line of the timelike and spacelike non developable ruled surfaces. Also, we have calculated the gaussian an…
Ruled surfaces with Ricci metrics use curves of constant torsion.
problem Characterizing ruled surfaces with Ricci metrics.
method Using curves of constant torsion to construct ruled surfaces.
result Helicoid is the only surface with constant mean curvature.
Study ruled surfaces with finite multiplicity, focusing on their curves and singularities.
problem Understanding ruled surfaces with finite multiplicity.
method Analyzing striction curves and singularities of ruled surfaces.
result Geometric meanings of invariants related to ruled surfaces.
Study ruled surfaces in 3D Riemannian manifolds, determining curvature and striction curves.
problem Characterize ruled surfaces in 3D Riemannian manifolds.
method Determine extrinsic and sectional curvature, introduce Sannia frame, characterize striction curves.
result Prove existence and uniqueness of striction curves in space forms, disprove in others.
Study on hypersurfaces with minimized distance between rulings.
problem Characterizing singularities and properties of two-ruled hypersurfaces.
method Characterization through striction curves and examination of pseudo-non-degenerate properties.
result Two-ruled hypersurfaces constructed from specific curves are pseudo-non-degenerate.
In this study, we define a family of ruled surfaces in the Euclidean 3-space E^3 and called similar ruled surfaces. We obtain some properties of these special surfaces and we show that developable ruled surfaces form a family of similar ruled surfaces if and only if the striction curves of the surfaces are similar curv…
Examples of foliations with infinite cohomology dimensions.
problem Understanding foliations with infinite dimensional cohomology.
method Presenting specific foliations and a general condition for their infinite cohomology.
result Foliations have infinite dimensional basic symplectic and complex cohomologies.
In this study, we define some new types of ruled surfaces called slant ruled surfaces. We give some characterizations for a regular ruled surface to be a slant ruled surface in Euclidean 3- space. We show that if the slant ruled surface is developable then the striction curve is a general helix or a slant helix accordi…
In this study, we consider the notion of similar ruled surface for timelike and spacelike ruled surfaces in Minkowski 3-space. We obtain some properties of these special surfaces in E_1^3 and we show that developable ruled surfaces in E_1^3 form a family of similar ruled surfaces if and only if the striction curves of …
Investigates billiard dynamics on smooth curves in higher dimensions.
problem Extending conventional billiard dynamics to higher-dimensional smooth curves.
method Area-preserving twist map, KAM theory, Mather's converse KAM, interpolating Hamiltonians.
result Uniform distribution of impact points for small chords on nice wires.
Classifies rank-one submanifolds in Euclidean space.
problem Classifying submanifolds with singularities.
method Associate degree to ruled submanifolds and analyze singularities.
result An open and dense subset of rank-one submanifolds is the union of cylindrical, conical, and tangent regions.
The study characterizes constant curvature manifolds using ruled surfaces.
problem Characterizing manifolds of constant curvature using ruled surfaces.
method Investigating ruled surfaces in 3d Riemannian manifolds, finding stiction curve, distribution parameter, and fundamental forms.
result Identifies necessary and sufficient conditions for extrinsically flat surfaces to be ruled and proves manifold properties.
Real line arrangements with 3n lines intersect each other in n+1 points are related to finite complex reflection groups.
problem Identifying all real line arrangements in CP^2 with the Hirzebruch property.
method Analyzing the intersections of lines and confirming the relationship with finite complex reflection groups.
result There exist exactly four real line arrangements in CP^2 satisfying the Hirzebruch property.
The paper describes the geometric properties of line congruences' singularities.
problem Understanding the singularities of generic line congruences.
method Use of an equiaffine pair to define generic line congruences.
result Geometric description of folds, cusps, and swallowtails as singularities of generic line congruences.
This paper generalizes the envelope of mid-lines to intermediate lines for a plane curve.
problem Understanding the envelope of intermediate lines for a plane curve.
method Using singularity theory techniques to analyze the local behavior of the envelope of intermediate lines.
result The envelope of intermediate lines (EIL) is formed by three disconnected sets: AEIL, the curve itself, and IPTL. Topology of space line arrangements depends on line count and multiple points.
problem Understanding the topology of space line arrangements.
method Determined by the number of lines and data on multiple points.
result Topology of complements is determined by line count and multiple points.
Study restricts line arrangements with odd points using topological arguments.
problem Restrictions on line arrangements with singular points of odd multiplicity.
method Topological arguments on locally-flat spheres in 4-manifolds.
result No line arrangement with 13 lines and only triple points exists.
A special group of transformations of the real line cannot act effectively on it.
problem Understanding the limitations of transformations on the real line.
method Analyzing the group of orientation-preserving quasi-isometries of the real line.
result The group of quasi-isometries of the real line cannot act effectively on the line.
This paper explores dual relations between line congruences and surfaces in 3D and 4D.
problem Understanding the duality between line congruences in R3 and surfaces in R4. method Analyzes the correspondence between principal lines and asymptotic lines, ridge curves and flat ridge curves, and subparabolic curves.
result Discusses the behavior of subparabolic curves at the discriminant curve of the line congruence and the parabolic curve of the dual surface.
Study Blaschke's asymptotic lines on surfaces in 3D space.
problem Characterize Blaschke's asymptotic lines on surfaces in 3D.
method Analyze binary differential equations near cusp and umbilic points.
result Describe Blaschke's asymptotic lines near Euclidean parabolic set.
Study on surfaces containing twistor lines, proving their density and bounds.
problem Understanding the distribution and maximum number of twistor lines on algebraic surfaces.
method Analyzing ideal sheaves, proving density in Grassmannian, and calculating bounds for surface degrees.
result Twistor lines are Zariski dense in the Grassmannian and have maximum bounds on their number.
Co-PLNet combines point and line predictions to improve wireframe parsing accuracy and efficiency.
problem Separate line and point predictions lead to inconsistent wireframes.
method Co-PLNet uses a Point-Line Prompt Encoder to convert early point detections into spatial prompts, which guide line refinement.
result Co-PLNet achieves better accuracy and robustness in wireframe parsing compared to existing methods.
Characterizes winding of braided vector fields in tubular domains.
problem Understanding the topology of braided vector fields in complex domains.
method Defines field line winding as a measure of entanglement, proving its uniqueness in classifying vector field topology.
result Field line winding uniquely classifies the topology of braided vector fields.
Classifies generic singularities of line fields on 2D manifolds.
problem Classifying singularities of line fields on 2D manifolds.
method Identifying line fields as bisectors of pairs of vector fields, considering singularities as zeros of vector fields, and using a natural topology in the space of pairs of vector fields.
result Generic singularities of line fields on 2D manifolds are topologically equivalent to the Lemon, Star, and Monstar singularities.
Affine maps on tori preserve lines.
problem Characterizing affine automorphisms on tori.
method Analogous to Euclidean space, established for tori.
result Affine maps on tori preserve lines.
Classifies metric lines in Engel-type groups, a step towards solving sub-Riemannian manifold problems.
problem Classifying metric lines in Engel-type groups.
method Sequence method to study metric lines in jet space.
result Classified metric lines of Engel-type groups $\Eng(n)$.
New Calabi-Yau metrics with conical singularities are created near complex lines.
problem Creating Calabi-Yau metrics with conical singularities near complex lines.
method Using branched covering arguments to construct metrics with conical singularities.
result Calabi-Yau metrics with unstable conical singularities are successfully created.
Constructs geodesic lines in curved surfaces using min-max methods.
problem Constructing geodesic lines in curved surfaces where minimization schemes fail.
method Employing min-max methods to construct uncountably many geometrically distinct geodesic lines.
result Proves the existence of uncountably many properly embedded geodesic lines with Morse index ≤ 1.
The paper examines asymptotic lines of plane fields in 3D space.
problem Qualitative properties of asymptotic lines in plane fields.
method Analysis of null directions and Gaussian curvature.
result Asymptotic lines coincide with classical ones in completely integrable fields.
Paper classifies maximal surfaces with planar curvature lines and explores their singularities.
problem Classifying maximal surfaces with specific curvature lines.
method Alternative method to refine classification and investigate singularities.
result Existence of a deformation consisting of all maximal surfaces with planar curvature lines.
The paper explores reflection principles for lightlike line segments on maximal surfaces.
problem Reflection property does not hold for lightlike line segments on maximal surfaces.
method Analyzes reflection properties for lightlike line segments connecting shrinking singularities.
result Shows a kind of reflection principle for lightlike line segments on maximal surfaces.
This paper studies topological properties of line arrangements in complex projective plane.
problem Understanding the topology of line arrangements in complex projective plane.
method Established foundational results using homological methods to compute cohomology rings and studied homotopy types.
result Complement of a symplectic line arrangement has the homotopy type of a minimal CW complex.
Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.
problem Discretizing surfaces with spherical curvature lines.
method Lie-geometric discretisation in terms of principal contact element nets.
result Circular nets with two families of spherical parameter lines are related to Darboux cyclides.
The paper explores graphons of line graphs from sparse finite graphs.
problem Estimating graph limits from sparse finite graphs.
method Mapping finite graphs to their line graphs and analyzing graphs with the square-degree property.
result Graphons of line graphs can distinguish between sparse graphs like star graphs and superlinear preferential attachment graphs.
Estimates cohomology dimensions for Nakano q-semipositive line bundles.
problem Estimating cohomology dimensions for Nakano q-semipositive line bundles.
method Asymptotic estimates for high tensor powers of semipositive line bundles over q-convex manifolds and various complex manifolds.
result Optimal order estimates for cohomology dimensions.
New method describes entanglement of straight lines in 3D space.
problem Tackles the geometry and topology of configurations of straight lines.
method Introduces direction matrices and a discrete motion principle.
result Shows n-crosses as links of pairwise connected unknots.
Study affine curvature lines on surfaces in 3D space.
problem Understanding the behavior of affine curvature lines on surfaces.
method Analyzing binary differential equations and topological models.
result Obtained topological models and generic behavior of affine curvature lines.
We define a pseudo-inverse for line graphs using linear integer programming.
problem Not all graphs have a corresponding root graph, making the line graph operation non-invertible.
method Propose a linear integer program to edit the smallest number of edges in the line graph to recover a root graph.
result The pseudo-inverse operation is well-behaved and works in practice as shown by empirical experiments.
Shows CM line bundles are ample on K-stable varieties.
problem Ensuring CM line bundles are ample on K-stable varieties.
method Analyzes CM line bundles on K-stable varieties and their families.
result CM line bundles are ample on K-stable varieties with maximal variation.
Study proves Hodge symmetry on Oeljeklaus-Toma manifolds with line bundles.
problem Hodge symmetry on complex manifolds with line bundles.
method Analyzes Dolbeault cohomology of Oeljeklaus-Toma manifolds with holomorphic line bundles.
result Proves Hodge symmetry and vanishing/non-vanishing of Dolbeault cohomology.
The paper finds two types of metric lines in curve spaces.
problem Classifying metric lines in jet spaces of curves.
method Established the existence of two families of metric lines in the 2-jet space of plane curves.
result Found precise criteria for identifying metric lines in sub-Riemannian geodesics.
Paper develops an algorithm for finding lines of curvature on 4D hypersurfaces.
problem Finding lines of curvature on parametric hypersurfaces in Euclidean 4-space.
method Develops an algorithm using the extended Darboux frame.
result Obtains curvatures of lines of curvature on hypersurfaces.
We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line …
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds …