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.
Generic singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In this paper we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with re…
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 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. 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. Our aim is to generalize the result that two generic complex line arrangements are equivalent. In fact for a line arrangement A we associate its defining polynomial, the product of a_ix+b_iy+c_i, so that A = (f=0). We prove that the defining polynomials of two generic line arrangements are, up to a small deformation, t…
Paper maps Hamiltonians and line elements in manifolds.
problem Mapping among generalized Hamiltonians and line elements.
method Constructing Calabi's Riemannian Line Elements and solving matrix Riccati equations.
result Analytical and exact solutions of mapping between manifolds.
Study stability thresholds of big line bundles, proving bounds and generalizing results.
problem Stability thresholds of big line bundles and their asymptotic behavior.
method Explicit bounds on error terms, using quasi-monomial valuations to compute stability thresholds.
result Proves Jin--Rubinstein--Tian's questions affirmatively.
The Enneper surface and helix surfaces are unique in their geometric properties.
problem Characterizing surfaces with specific geometric properties.
method Analyzing isogonal lines and pseudo-geodesic lines in 3D Euclidean space.
result Helix surfaces and Enneper surface are the only surfaces with isogonal lines as generalized helices and pseudo-geodesic lines.
The paper studies how points and lines can move while preserving incidences.
problem Understanding how point-line configurations can move while maintaining their geometric relationships.
method Developed a projective rigidity matrix to analyze the infinitesimal motions and dependencies of point-line configurations.
result The symmetry-adapted projective rigidity matrix provides a more detailed analysis of symmetric configurations and their motions.
Generalizes cohomology ring result for combinatorial line arrangements.
problem Cohomology ring of boundary manifold for combinatorial line arrangements.
method Introduced boundary manifold, constructed homology cycles, computed cohomology ring.
result Cohomology ring of boundary manifold is isomorphic to double of Orlik-Solomon algebra.
This article is devoted to the study of cyclides osculating general surfaces. We show that generically, at any point of a surface, one has a one-parameter family of cyclides tangent to a surface curve of order three and among them just one is tangent to this curve of order four. This one will be called the osculating c…
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.
New discretizations of principal curvature lines discovered.
problem Discretizing principal curvature line parametrizations.
method Generalization of polar pairs of line congruences in the Lie quadric.
result New discretizations of orthogonal and Gauss-orthogonal parametrizations.
We show that a generic real projective n-dimensional hypersurface of degree 2n-1 contains "many" real lines, namely, not less than (2n-1)!!, which is approximately the square root of the number of complex lines. This estimate is based on the interpretation of a suitable signed count of the lines as the Euler number of …
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 on straight-line flows for generative modeling with theoretical obstructions.
problem Existence and obstructions of straight-line flows in generative modeling.
method Characterizations of straight-line flows through PDEs involving conditional statistics of stochastic processes.
result Sharp dichotomy in the existence of straight-line flows for targets with well-separated modes.
Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.
problem Non-semi-positivity of line bundles
method Introduce generalized Ueda obstruction classes and use Dolbeault resolution
result Recover classical examples and provide new examples of nef but not semi-positive line bundles
Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but non-isomorphic fundamental groups. To do so, the Alexander Invariant and certain invariants of combinatorial line arrangements a…
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.
There is a natural duality between line congruences in R3 and surfaces in R4 that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat r…
We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we…
Study weak geodesic lines in Kähler metric space, disproving a conjecture.
problem Disproving a conjecture about weak geodesic lines in Kähler metrics.
method Establish Ross-Witt Nyström correspondence, construct weak geodesic lines.
result Some weak geodesic lines are smooth, disproving a popular conjecture.
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 studies surfaces with spherical curvature lines and their generation by constrained elastic curves.
problem Understanding surfaces with spherical curvature lines and their generation mechanisms.
method The approach involves Lie sphere transformations, Legendre curves, and polynomial conserved quantities of connections.
result Lie applicable surfaces with exactly one family of spherical curvature lines are generated by the lift of constrained elastic curves.
In this work we study the affine principal lines of surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic and hyperbolic). We also describe the generic behavior of affine curvature lines …
Isothermic nets created from special maps for smooth surfaces.
problem Creating discrete curvature lines on surfaces.
method Special discrete holomorphic maps and lifted-folding.
result Isothermic nets with spherical parameter lines constructed efficiently.
The study generalizes a specific geometric correspondence to higher dimensions.
problem Understanding nondegenerate lines on holomorphic contact manifolds.
method Analyzing nondegenerate lines and corresponding distributions on higher-dimensional manifolds.
result A generalization of the (2,3,5)-distributions to higher dimensions. Study complex lines in symplectic geometry, generalizing previous results.
problem Understanding symplectic aspects of complex lines and their associated currents.
method Systematic study using Ahlfors currents, generalizing previous results.
result Ahlfors currents control the asymptotic behavior of pseudoholomorphic curves, showing convexity of the space of currents.
The paper explores affine geometry of line congruences using singularity theory.
problem Understanding the affine geometry of line congruences and their focal sets.
method Use of singularity theory to describe generic phenomena and singularities.
result Identification of a key projective quadric in the tangent space.
Six quaternionic lines with optimal angles found in 2D quaternion space.
problem Finding optimal configurations of quaternionic lines in 2D space.
method Simple presentation of lines as orbit of a reflection group, finding other optimal designs.
result Optimal spherical designs of 10, 15, and 20 lines in quaternion space.
Paper generalizes Hamiltonian mechanics using line bundles.
problem Mathematical foundations of measurand and units of measurement.
method Introduces line bundles over smooth manifolds as configuration spaces.
result Generalization successfully incorporates physical dimension and units.
Invariant Kähler metrics on line bundles are derived from the Calabi ansatz.
problem Finding invariant scalar-flat Kähler metrics on line bundles over generalized flag varieties.
method Proved using the Calabi ansatz and uniqueness in each Kähler class.
result Existence of a unique scalar-flat Kähler metric in each Kähler class.
In this paper we show that to each planar line arrangement defined over the real numbers, for which no two lines are parallel, one can write down a corresponding relation on Dehn twists that can be read off from the combinatorics and relative locations of intersections. This leads to an alternate proof of Wajnryb's gen…
The paper explores the geometry of level lines of quasiperiodic functions with many periods.
problem Describing the geometry of level lines of quasi-periodic functions with a large number of periods.
method Generalizes the Novikov problem to the multidimensional case of quasiperiodic functions.
result Arises of open or closed level lines of arbitrarily large sizes.
Study the monodromy and center-focus problems for rational maps defined by products of generic lines.
problem Monodromy and center-focus problems for rational maps defined by products of generic lines.
method Analyze the 1-homology group and meromorphic 1-forms to characterize vanishing Abelian integrals.
result Characterize meromorphic 1-forms whose Abelian integrals vanish on cycles around a center singularity.
Study on complex line fields on almost-complex manifolds, proving existence conditions.
problem Existence of linearly independent complex line fields on almost-complex manifolds.
method Prove necessary and sufficient conditions for the existence of one, two, or three fields over certain manifolds.
result Necessary and sufficient condition for the existence of complex line fields over certain manifolds.
We prove that every right-angled Artin group embeds into the C∞ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG group, embeds into the C∞ diffeomorphism group of the real line.
Let X be a complex-projective contact manifold whose second Betti-number is one. It has long been conjectured that X should then be rational-homogeneous, or equivalently, that there exists an embedding of X into a projective space whose image contains lines. Using methods introduced in math.AG/0206193, we show that X i…
Wilson lines generate positive Laurent polynomials in decorated triangulations.
problem Wilson lines and their coefficients in function algebras.
method Study of Wilson lines on marked surfaces and their matrix coefficients in function algebras.
result Matrix coefficients of Wilson lines give Laurent polynomials with positive integral coefficients.
Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.
problem Proving the existence of smooth solutions for Demailly's system.
method Used Demailly's system and Leray-Schauder degree theory to reduce the problem.
result Proved existence of smooth solutions for direct sums of ample line bundles.
Identifying a potentially large number of simultaneous line outages in power transmission networks in real time is a computationally hard problem. This is because the number of hypotheses grows exponentially with the network size. A new "Learning-to-Infer" method is developed for efficient inference of every line statu…
A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical properties, e.g. the principal curvature directions dynamics on surfaces or the stress…
We prove that a finite type curve is an ξ-asymptotic line (without parabolic points) of a suitable plane field. It is also given an explicit example of a hyperbolic closed finite type ξ-asymptotic line. These results obtained here are generalizations, for plane fields, of the results of V. Arnold [4].
Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular,…
The study examines conditions that prevent null geodesic lines in spacetimes, impacting cosmological geometry.
problem Preventing the existence of null geodesic lines in spacetimes.
method Identifying geometric conditions on foliations of spacetimes that prevent null geodesic lines, especially for spacetimes with compact Cauchy hypersurfaces.
result Conditions on foliations can prevent null geodesic lines, leading to restrictions on cosmological spacetime geometry.
Let X be a complex Fano-manifolds with second Betti-number 1 which carries a contact structure. It follows from previous work that such a manifold can always be covered by lines. Thus, it seems natural to consider the geometry of lines in greater detail. In this brief note we show that if x in X is a general point, the…