Study on defect topology in smectic liquid crystals.
problem Topological classification and combination rules of smectic defects.
method Topological analysis of point defects and disclination lines in 2D and 3D smectic liquid crystals.
result Combination rules for smectic point defects in 2D and 3D, showing path dependence.
Study on materials with disclinations, limiting their size.
problem Limiting the size of disclinations in materials with symmetries.
method Defining material-uniform hyperelastic bodies with disclinations, rigorously analyzing their properties.
result The size of disclinations is limited by the symmetries of the constitutive relation.
The homotopy theory of topological defects in ordered media fails to completely characterize systems with broken translational symmetry. We argue that the problem can be understood in terms of the lack of rotational Goldstone modes in such systems and provide an alternate approach that correctly accounts for the intera…
Generalizes beam models to include curvature and torsion.
problem Modeling curvature and torsion in unidimensional structures.
method Generalized Euler-Bernoulli and Timoshenko beam models using non-holonomic kinematics.
result Generalized beams exhibit curvature and torsion, with torsion in Timoshenko model.
Study of knotted defects in smectic liquid crystals using topological knot theory.
problem Understanding the topological structure of knotted defects in smectic liquid crystals.
method Investigation of screw and edge dislocations, focusing on their radial surface structure and knot fibration.
result Established a connection between smectic defects and knot theory, revealing the topological knotting of defects.
Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with a configuration of a body in classical elasticity is the sum of local contributions …
A new model for defective media using two scales.
problem Modeling defects in media with two scales.
method Generalization of Riemann-Cartan manifolds and fibre bundle theory, constructing a first-order placement map.
result Emergent behaviors like dislocations and disclinations arise from the interaction of macroscopic and microscopic scales.
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.
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 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.
A line arrangement of 3n lines in CP2 satisfies Hirzebruch property if each line intersect others in n+1 points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in CP2 is…
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.
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.
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.
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.
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.
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 …
Geometric quantization extended to big line bundles.
problem Quantization of line bundles with large curvature.
method Proving asymptotic isometry and submultiplicative norms equivalence, showing Mabuchi geodesic rays.
result Bounded submultiplicative filtrations on big line bundles lead to Mabuchi geodesic rays.
Line graph transformation aids graph isomorphism tests by excluding challenging graph properties.
problem Limited theoretical understanding of line graph transformation's impact on GNN models.
method Examined CFI and strongly regular graphs, showing line graph transformation helps WL tests distinguish these graphs.
result Line graph transformation aids WL tests in distinguishing challenging graph properties.
New BDEs reveal singular surfaces from line congruences.
problem Understanding binary differential equations associated with line congruences.
method Applied pointwise to quadratic differential forms, studying quotients of quadratic forms and associated polar lines.
result Introduced a new singular surface in Euclidean 3-space.
We give quantitative and qualitative results on the family of surfaces in CP3 containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines E. We prove that its general element is a smooth surface containing E and no other line. Afterwards we prove that …
Study conic-line arrangements of degree 7, finding their topology and components.
problem Understanding the topology and components of conic-line arrangements of degree 7.
method Examined the embedded topology using SageMath and Sirocco package.
result Determined the number of connected components and fundamental groups.
The paper estimates the dimension of cohomology for semipositive line bundles on various manifolds.
problem Estimating the dimension of cohomology for semipositive line bundles over different types of manifolds.
method Asymptotic estimates for harmonic (0,q)-forms with high tensor powers of semipositive line bundles. result Asymptotic estimates for the dimension of cohomology of semipositive line bundles on various manifolds.
Finite type curves are asymptotic lines of plane fields, with examples.
problem Characterizing finite type curves as asymptotic lines of plane fields.
method Proving finite type curves are asymptotic lines of suitable plane fields, providing examples.
result Generalizations of V. Arnold's results for plane fields.
We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link configurations can be realized by complex line arrangements but not by real line arrangem…
The paper classifies singularities of line congruences in 4D space.
problem Classifying singularities of line congruences in 4D space.
method Generic classification approach for 3-parameter line congruences and Blaschke normal congruences.
result Generic classification of singularities of 3-parameter line congruences in R4. Study differential invariants for line bundle operators.
problem Conditions for equivalence of differential operators in line bundles.
method Use differential invariants and groups of automorphisms.
result Find conditions for equivalence of differential operators.
KQSP method prevents crossing issues in probabilistic K-line forecasts.
problem Quantile and K-line crossing issues in probabilistic K-line forecasts.
method Parameter-free and training-free reconciliation method (KQSP).
result KQSP reduces crossing rates to zero for all test data.
Study three discrete envelope types of polygon bisection lines.
problem Understanding different envelope types of polygon bisection lines.
method Examined three distinct notions of discrete envelopes.
result Connected three different notions of discrete envelopes.
Paper proves ε-regularity for line bundle mean curvature flow.
problem Proving regularity for a specific type of geometric flow.
method Develops a scale-invariant monotone quantity and defines self-shrinkers.
result Establishes ε-regularity theorem for line bundle mean curvature flow.
New framework for analyzing line fields on surfaces, proving stability under specific conditions.
problem Understanding structural stability and generic transitions of line fields on surfaces.
method Developed a new topological framework and introduced representations of complete invariants for line fields and their transitions.
result Line fields with 1-prong and 3-prong singularities are generic under an incompressibility condition.
The Samuelson condition is not satisfied by tangent lines of quadratic curves.
problem Area condition for Lagrangian 2-web
method Show that the Samuelson condition is not satisfied
result The Samuelson condition is not satisfied by tangent lines of quadratic curves.