Self-affine arcs without inner weak separation are parabolic segments.
problem Characterizing self-affine Jordan arcs without parabolic segments.
method Analyzing the weak separation property and proving implications for arc types.
result Self-affine Jordan arcs without parabolic segments are attractors of multizippers.
Study properties of self-similar continua with finite intersection property.
problem Characterize self-similar continua with finite intersection property.
method Prove intersection graph criterion, finite order theorem, and parameter matching theorem.
result All Jordan arcs starting from a intersection point in such continuum on a plane should have the same slope parameter at that point.
Study shows singular set of distance functions is delta-convex.
problem Understanding singular set of distance functions in Finsler manifolds.
method Proved singular set is delta-convex hypersurfaces or Jordan arcs up to exceptional sets.
result Optimal results in Finsler manifolds, even in Euclidean space.
The paper finds a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.
problem Finding a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.
method Defining a complex projective structure and using accessory parameters to characterize the curve.
result The accessory parameters are the residues of the quadratic differential comparing the projective structure to the trivial one.
Square-like quadrilaterals inscribed in space curves proven for finite total curvature.
problem Square-peg problem in space curves.
method Local curvature analysis and limiting argument on approximating curves.
result Every embedded curve with finite total curvature has an inscribed square-like quadrilateral.
Let R be an open Riemann surface. In this paper we prove that every continuous function M→Rn, n≥3, defined on a divergent Jordan arc M⊂R can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theor…
We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type A nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelf…
We characterize the differentiable points of the distance function from a closed subset N of an arbitrary dimensional Finsler manifold in terms of the number of N-segments. In the case of a 2-dimensional Finsler manifold, we prove the structure theorem of the cut locus of a closed subset N, namely that it is a lo…
Geometric inequalities for equipotential curves derived from convex entropy.
problem Geometric relations for equipotential curves defined by harmonic functions.
method Constructing an entropy for each level set and proving convexity.
result Geometric inequalities for curvature and gradient magnitude on equipotential curves.
The paper studies curvatures in metric Jordan algebras, proving unique connections and curvature formulas.
problem Investigating curvatures in metric Jordan algebras.
method Defined the Jordan-Levi-Civita connection, introduced curvature tensors, and proved curvature formulas.
result Every formally real Jordan algebra admits a metric of non-positive Jordan curvature and a Jordan-Einstein metric of negative Jordan scalar curvature.
Study pseudo-Riemannian metrics on Jordan superalgebras.
problem No specific problem stated; focus on metrics.
method Coadjoint orbit-like construction for pseudo-Euclidean Jordan superalgebras.
result Investigated canonical pseudo-Riemannian metrics.
We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which app…
Researchers solve the realization of Jordan-Kronecker invariants in Lie algebras.
problem Identifying which Jordan-Kronecker invariants can be realized by Lie algebras.
method Analyzing the Kronecker and Jordan cases, proving impossibility for certain invariants, and describing realizability for others.
result Complete solution for Jordan and Kronecker cases, partial answers for others.
New Jordan Floer homology shows every curve inscribes every isosceles trapezoid.
problem Understanding inscriptions of isosceles trapezoids in Jordan curves.
method Constructing a new Lagrangian Floer homology chain complex.
result Establishes new cases of non-smooth Jordan curves inscribing isosceles trapezoids.
Every curve can fit countless rhombuses.
problem Finding many rhombi within any curve.
method No curve regularity assumed.
result Uncountably many rhombi fit every curve.
We present new results about Jordan algebras and Jordan coalgebras, and we discuss about their connections with the Yang-Baxter equations.
We study a notion of "width" for Jordan curves in CP1, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three-space. A similar invariant in the setting of anti de Sitter geometry was used by Bonsante-Schle…
Selects points from Jordan domains on Riemannian surfaces.
problem Selecting points from Jordan domains in Riemannian surfaces.
method Fiber bundle theory and conformal mappings.
result Space of Jordan domains retracts onto round disks.
Let s be at least 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s,s) which are not locally homogeneous but whose curvature tensors never the less exhibit a number of important symmetry properties. They are curvature homogeneous; their curvature tensor is modeled on that of a local symmetric spac…
Jordan algebras in information geometry linked to metrics on probability distributions.
problem Understanding Jordan algebras in information geometry.
method Inspired by Kirillov's coadjoint orbits, a pseudo-Riemannian metric is constructed on Jordan algebra leaves.
result Not all points in the dual space lie on a leaf, and the metric structure depends on the cone of positive functionals.
Square inscribed in a curve made of two graph functions.
problem Finding inscribed squares in curves formed by graph functions.
method Analysis of spectral invariants of Jordan Floer homology under curve perturbations.
result Existence of inscribed squares in curves with specific Lipschitz constants.
Paper extends theorem on covering spaces and Jordan curves.
problem Covering and extending theorems for Alexandrov spaces.
method Introduces proximal homotopic cycles to extend the Mitsuishi-Yamaguchi theorem.
result Extensions of the Mitsuishi-Yamaguchi Good Covering Theorem and Jordan curve theorem.
In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in H2×R. As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary C is a Jordan curve homologous to zero in the asymptotic boundary of H2×R, say $\partial_\infty H^2\tim…
We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic groups (not necessarily affine) over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan.
To study the Lawson-Osserman's counterexample to the Bernstein problem for minimal submanifolds of higher codimension, a new geometric concept, submanifolds in Euclidean space with constant Jordan angles(CJA), is introduced. By exploring the second fundamental form of submanifolds with CJA, we can characterize the Laws…
For a Jordan domain in the plane the length metric space of points connected to an interior point by a curve of finite length is a CAT(0)space and Gromov hyperbolic. With respect to the cone topology, that space plus its boundary at infinity is topologically the same as the original Jordan domain.
In this paper, we formulate a new local move on virtual knot diagram, called arc shift move. Further, we extend it to another local move called region arc shift defined on a region of a virtual knot diagram. We establish that these arc shift and region arc shift moves are unknotting operations by showing that any virtu…
Two curves in hyperbolic space share a bounded distance, leading to a minimizing surface.
problem Finding a surface minimizing area between two disjoint curves in hyperbolic space.
method Analyzing the asymptotic boundary of hyperbolic 3-space, applying Definition 1.8 for distance bounds, and proving Theorems 1.7 and 1.11.
result Existence of an area-minimizing surface between two disjoint curves with bounded distance.
The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up-dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iordanescu [364w], where sugestions for further developments ar…
Two proofs show that removing a loop from a plane circuit splits the plane.
problem Proving the Weak Jordan Theorem about plane circuits.
method Detailed presentation of Thomassen's and Filippov's proofs.
result The complement of any loop in a plane circuit is disconnected.
Pseudo-Riemannian manifolds of balanced signature which are both spacelike and timelike Jordan Osserman nilpotent of order 2 and of order 3 have been constructed previously. In this short note, we shall construct pseudo-Riemannian manifolds of signature (2s,s) for any s (which is at least 2) which are spacelike Jordan …
Study finds conditions for operator fields to be in strictly upper triangular form in small dimensions.
problem Jordan-Chevalley decomposition for operator fields in small dimensions.
method Tensorial conditions and proof of conjecture for higher order brackets.
result Proves Tempesta-Tondo conjecture for higher order brackets.
Study on unknotting twisted knots using arc shift and region arc shift moves.
problem Unknotting twisted knots and finding bounds for region arc shift number.
method Introduced arc shift move and region arc shift move for twisted knots.
result Found families of twisted knots with specific arc shift and region arc shift numbers.
We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type (r,s) in a vector space of signature (p,q). We then use these examples to establish some results concerning higher order Osserman and highe…
It is shown that the projection image of an oriented spatial arc to any oriented plane is approximated by a unique arc diagram (up to isomorphic arc diagrams) determined from the spatial arc and the projection. In a separated paper, the knotting probability of an arc diagram is defined as an invariant under isomorphic …
The paper classifies hypercomplex Lie algebras and solvmanifolds using quaternionic Jordan form.
problem Classifying hypercomplex Lie algebras and solvmanifolds.
method Application of quaternionic Jordan form to Lie algebras and solvmanifolds.
result Infinitely many hypercomplex almost abelian solvmanifolds constructed.
Similarity maps cyclic quadrilaterals onto smooth curves.
problem Mapping cyclic quadrilaterals onto smooth Jordan curves.
method Uses the theorem of Polterovich and Viterbo.
result Existence of orientation-preserving similarity.
Automorphism groups of Hopf manifolds are finite and have a bounded order.
problem Understanding the structure of automorphism groups of Hopf manifolds.
method Proved that the automorphism groups of Hopf manifolds are Jordan.
result Automorphism groups of Hopf manifolds are finite and have a bounded order.
NT probability measures knotting in 3D arc systems.
problem Measuring knotting in 3D arc systems.
method Transforming polygonal arcs into unique diagrams, generalizing NT probability.
result Properties of NT probability for 3D arc systems are shown.
Non-trivialization probability of arc system in 3D space
problem Defining and generalizing the knotting probability of an arc diagram in 3D space
method Transforming polygonal arcs in 3D space into unique arc diagrams
result Introducing and generalizing the Non-Trivialization probability (NT probability) for arc systems in 3D space
The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.
problem Analyzing the geometric and combinatorial effects of smoothing intersections in arcs or curves.
method Geometric and combinatorial analysis, proving tautness and arc length spectrum properties.
result Shortest arcs with self-intersections have exactly or at most one more self-intersection than the self-intersection number.
Curves inscribe rectangles with positive area.
problem Finding angles for inscribing rectangles within Jordan curves.
method Proving existence of a subset of angles with measure at least A/R^2.
result Angles inscribing rectangles have measure at least A/R^2.
Minimal grid diagrams for 15,735 knots with 14 crossings and arc index 14.
problem Representing prime knots with 14 crossings and specific arc indices using grid diagrams.
method Enumerated all prime knots with 14 crossings, categorized by arc index, and found minimal grid diagrams for those with arc index 14.
result 8,027 knots with arc index 13 and 15,735 knots with arc index 14 were represented by minimal grid diagrams.
This paper calculates stick numbers for rail arcs and knot classes.
problem Calculating the minimum number of sticks needed for rail arcs and knot classes.
method Rail isotopies, ambient isotopies, winding number invariant, and lattice stick number.
result Calculates stick numbers for rail arcs and knot classes with crossing number at most 9.
Counts arcs in surfaces, proving convergence of geodesic currents.
problem Counting arcs of the same type in compact surfaces and related geometries.
method Derives convergence of geodesic currents to prove arc counts.
result Proves convergence of geodesic currents, leading to arc counting results.
Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.
problem Understanding geometric structures related to geodesic foliations and dynamics.
method Introducing symmetric Poisson structures, proving correspondences with geodesic foliations and Jordan algebras.
result Symmetric Poisson structures correspond to totally geodesic foliations and Jacobi-Jordan algebras.
Let C be a real-analytic Jordan curve in R3. Then C cannot bound infinitely many disk-type minimal surfaces which provide relative minima of area.
Study arcs on surfaces, focusing on topological aspects and group actions.
problem Understanding arcs and their complements on surfaces.
method Characterize infinite-type surfaces via homeomorphic subsurfaces, construct actions on arc graphs.
result New characterisation of infinite-type surfaces and actions on arc graphs.