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.
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.
Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.
problem Understanding real lines on real del Pezzo surfaces of degree 1.
method Intrinsically defined Pin-structure on the real locus.
result The difference between hyperbolic and elliptic lines is always 16.
Classifies Real line bundles with Real connections on manifolds with involution.
problem Classifying Real line bundles with Real connections on manifolds with involution.
method Defines Real smooth Deligne cohomology to interpolate between equivariant sheaf cohomology and smooth imaginary-valued forms.
result Classifies Real line bundles with Real connections on manifolds with involution.
Study real line subbundles on curves, extending classical work.
problem Understanding real line subbundles in real bundles on curves.
method Application of Atiyah's techniques and work of Lange-Narasimhan.
result Describes the Galois action on the set of lines through a real point in the moduli space of such bundles.
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 …
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.
Constructs a topological cover of real line's multiplicative group.
problem Topological cover of real line's multiplicative group.
method Homological algebra, 2D Lorentz geometry, high-school trigonometry.
result Interesting topological cover constructed.
Study real Mordell-Weil group and real lines on rational elliptic surfaces and del Pezzo surfaces.
problem Characterize the real Mordell-Weil group and real lines on rational elliptic surfaces and del Pezzo surfaces.
method Explicit description of isotopy types of real lines and presentation of MW group in mapping class group.
result Explicit formula for the action of MW group in H1(XR).
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.
Proves non-properness set of 3D polynomial homeos can't be a line.
problem Non-properness set of 3D polynomial homeomorphisms.
method Proof by contradiction, using topological properties.
result Non-properness set cannot be homeomorphic to the real line.
We explore the connection between Hilbertian metrics and positive definite kernels on the real line. In particular, we look at a well-known characterization of translation invariant Hilbertian metrics on the real line by von Neumann and Schoenberg (1941). Using this result we are able to give an alternate proof of Boch…
New topological invariant distinguishes real line arrangements with same combinatorics.
problem Determining topological properties from combinatorial data of real line arrangements.
method Introducing chamber weight invariant based on dual configuration points in real projective plane.
result New Zariski pairs of 13, 15, and 17 lines with different topological embeddings.
Efficient method identifies multiple power grid outages in real-time.
problem Identifying simultaneous line outages in large power networks is computationally challenging.
method Developed a 'Learning-to-Infer' method to efficiently infer every line status.
result The method achieves excellent performance in identifying multi-line outages in real-time with minimal labeled data.
The center of a quotient group of piecewise linear homeomorphisms is trivial.
problem Understanding the structure of a specific group of homeomorphisms.
method Analyzing a quotient of piecewise linear homeomorphisms of the real line.
result The center of the quotient group is trivial.
We prove that the group D^r(R) of C^r diffeomorphisms of the real line, endowed with the compact-open and Whitney C^r topologies, is bihomeomorphic to the group H(R) of homeomorphisms of the real line endowed with the compact-open and Whitney topologies. This implies that the diffeomorphism group D^r(R) endowed with th…
Classifies and constructs intertwining differential operators between line and vector bundles over real projective space.
problem Classifying and constructing intertwining differential operators between line and vector bundles over real projective space.
method F-method for classification and construction of intertwining differential operators.
result Generalizes a classical result of Bol for SL(2,R) and classifies intertwining operators for SL(n,R). The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.
problem Classifying and constructing differential symmetry breaking operators.
method Utilizing factorization identities and branching laws of generalized Verma modules.
result Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces are classified and constructed.
Algorithm finds real line mapping from points under ordinal constraints.
problem Finding a mapping from points to real line under ordinal constraints.
method Approximation algorithm for dense case in O(n7)+(1/ε)O(1/ε1/8)n time. result Computes a solution satisfying (1−O(ε1/8))-fraction of all constraints. This study classifies metric lines in jet space.
problem Classifying metric lines in jet space.
method Using an intermediate 3D sub-Riemannian space to prove the main theorems.
result Partial results on the classification of metric lines in Jk(R,R). Study shows fundamental group and lower central series torsion are not determined by intersection lattices of real line arrangements.
problem Determining fundamental groups and lower central series quotients of real line arrangements by their intersection lattices.
method Computer-assisted proofs and deducing from known results.
result Fundamental groups and lower central series quotients are not combinatorially determined.
We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic involution on the threefolds. Our threefolds are birational to double coverings of CP…
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.
The study defines invariants for time-like surfaces with real asymptotic lines.
problem Characterizing time-like surfaces with real asymptotic lines.
method Fundamental theorem of Bonnet-type, canonical parameters, invariant functions, PDEs.
result Time-like surfaces are determined by four invariant functions, two of which can be Gauss and mean curvature.
The paper explores how configurations of lines can be realized in different geometric settings.
problem Whether configurations of lines can be realized by spheres in various geometric settings.
method Investigates realizability of configurations in topological, symplectic, and smooth categories in the complex projective plane.
result Obstructions to realizability in the topological category for configurations specified by projective planes over finite 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…
Complete duality theory for martingale optimal transport on the line.
problem Optimal transport between two probability measures on the real line.
method Quasi-sure formulation of the dual problem, showing complete duality theory for general marginals and measurable reward functions.
result Absence of duality gap and existence of dual optimizers for general marginals and measurable reward functions.
The center of the group of quasi-isometries of the real line is trivial.
problem Identifying the center of the group of quasi-isometries of the real line.
method Analyzing the group structure and using the quasi-isometry properties to show the triviality of the center.
result The center of the group of quasi-isometries of the real line is trivial.
We give local, explicit representation formulas for n-dimensional spacelike submanifolds which are marginally trapped in the Minkowski space, the de Sitter and anti de Sitter spaces and the Lorentzian products of the sphere and the hyperbolic space by the real line.
In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R^3. By means of a connection of the differential equations for these curvature lines and real Ricc…
We discuss holomorphic isometric embeddings of the projective line into quadrics using a generalisation of the theorem of do Carmo--Wallach to provide a description of their moduli spaces up to image and gauge--equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.
We construct real analytic flat Moebius strips of arbitrary isotopy types, whose centerlines are geodesics or lines of curvature.
Calculations of Orlicz cohomology for simple manifolds and related inequalities.
problem Understanding Orlicz cohomology and inequalities for basic manifolds.
method Calculations of Orlicz cohomology for real line, hyperbolic plane, and ball; discussion of inequalities.
result Relationship between Orlicz cohomology and Poincaré--Sobolev--Orlicz inequalities.
The present paper is a contribution to categorial index theory. Its main result is the calculation of the Pfaffian line bundle of a certain family of real Dirac operators as an object in the category of line bundles. Furthermore, it is shown how string structures give rise to trivialisations of that Pfaffian.
We study Milnor fibers of complexified real line arrangements. We give a new algorithm computing monodromy eigenspaces of the first cohomology. The algorithm is based on the description of minimal CW-complexes homotopic to the complements, and uses the real figure, that is, the adjacency relations of chambers. It enabl…
Study of moduli spaces for special harmonic maps from projective line to quadrics.
problem Characterizing moduli spaces of Einstein-Hermitian harmonic maps.
method Using vector bundles and representation theory, the authors describe the moduli spaces of these maps.
result The dimension of the moduli spaces is independent of the Einstein-Hermitian constant.
BOIDS optimizes high-dimensional problems by guiding optimization with one-dimensional lines.
problem Scaling Bayesian Optimization to high-dimensional problems.
method BOIDS uses a sequence of one-dimensional direction lines guided by an adaptive selection technique and incorporates subspace embedding for efficiency.
result BOIDS outperforms state-of-the-art methods on various synthetic and real-world problems.
Describes curvature lines on a double torus in 4D space.
problem Analyzing curvature lines on a complex geometric shape.
method Using polynomial gradient and Milnor fibration to define curvature lines, then projecting them into 3D space.
result Complete description of curvature lines on the double torus.
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 $\Eng(n)$.
Paper describes an on-line PCA algorithm for real-time data analysis.
problem Continuity of principal components during real-time analysis.
method Incremental PCA algorithm that updates coefficients on-line without keeping all samples.
result Algorithm maintains the same transformation coefficients as the batch version.
Study of twistor spaces and minitwistor spaces for ALE gravitational instantons.
problem Characterizing the geometry of ALE gravitational instantons of type Amodd. method Analyzing the base locus of linear systems and using distinguished twistor lines.
result Explicit determination of images of certain twistor lines and description of real minitwistor lines.
We compute the fundamental groups of the complements of the family of real conic-line arrangements with up to two conics which are tangent to each other at two points, with an arbitrary number of tangent lines to both conics. All the resulting groups turn out to be big.
Classifies Real primary Hopf surfaces and their associated groups.
problem Classifying Real primary Hopf surfaces and their associated groups.
method Complete classification up to Real biholomorphisms and equivariant diffeomorphisms.
result Detailed description of groups associated with Real primary Hopf surfaces.
Isothermic tori with one planar curvature line found and characterized.
problem Classifying isothermic tori with specific curvature lines.
method Complex analytic methods and explicit theta function formulas.
result Explicit formulas for family of plane curves and their relation to hyperbolic elastica.
The notion of max-plus convex subset of Euclidean space can be naturally extended to other linear spaces. The aim of this paper is to describe the topology of hyperspaces of max-plus convex subsets of Tychonov powers Rτ of the real line. We show that the corresponding spaces are AR's if and only if τ≤ω1…
The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description …
We give a new algorithm computing local system cohomology groups for complexified real line arrangements. Using it, we obtain several conditions for the first local system cohomology to vanish and to be at most one-dimensional, which generalize a result by Cohen-Dimca-Orlik. The conditions are described in terms of dis…
The study characterizes elements of a group quotient and finds a non-locally indicable left-orderable group.
problem Understanding the structure and properties of a specific group quotient.
method Introduced an invariant for quasi-isometries of the positive real line and split it into units.
result Found a quotient of the quasi-isometry group of the positive real line that is left-orderable but not locally indicable.