Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact…
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48 results for “tangency points”
Neural Point-Formscs.LG
Introduces neural point-forms for learning geometric features from noisy point clouds.
problem Learning geometric features from noisy point clouds with missing tangency information.
method Uses Laplacian-based techniques to build comparison matrices for point clouds, proving consistency under various assumptions.
result Neural point-forms provide a competitive and interpretable representation, especially beneficial for dense or manifold-like structures.
Solves Apollonius' problem using oriented circles and inversive geometry.
problem Constructing a circle tangent to three given circles.
method Using oriented circles and inversive invariants, reversing each given circle to find solutions.
result The problem has 0, 1, or 2 solutions, depending on the configuration of given circles.
Paper introduces new invariant for pairs of immersions.
problem Understanding behavior of immersions through tangencies and triple points.
method Introduces -invariant for oriented pairs of immersions, invariant under inverse tangencies and triple points.
result Invariant changes under direct tangencies but remains invariant under orientation change and inverse tangencies.
Research shows quadratic growth in derivative maxima for certain interval diffeos with parabolic fixed points.
problem Analyzing the growth of derivative maxima for interval diffeomorphisms with parabolic fixed points.
method Examining diffeomorphisms with only parabolic fixed points, focusing on tangency and repelling behavior.
result Maximal growth of derivative maxima is exactly quadratic for diffeomorphisms with a non-quadratic tangency to identity at a repelling fixed point.
Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.
problem Determining the domain of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds with tangency points.
method Using tools from Lie groupoids, natural domains of perturbations are found.
result Method allows treatment of geometries with tangency points.
Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of point…
We study the order of tangency between two manifolds of same dimension and give that notion three quite different geometric interpretations. Related aspects of the order of tangency, e.g., regular separation exponents, are also discussed.
Proposes a new model to maximize out-of-sample Sharpe ratios by forecasting tangency portfolios.
problem Maximizing Sharpe ratios when returns and covariances are not stationary.
method Forecast the tangency portfolio using vector autoregressions and invest in the minimum Euclidean distance portfolio.
result Empirically validated superior out-of-sample Sharpe ratios.
In this paper we study the affine geometric structure of the graph of a polynomial . We provide certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is hyperbolic or elliptic. We analyse the extension to the real projective plane of…
We show that for a residual subset of diffeomorphisms far away from tangency, every non-trivial chain recurrent class that is accumulated by sources ia a homoclinic class contains periodic points with index 1 and it's the Hausdorff limit of a family of sources.
We consider smooth 1-parameter families of plane curves tangent to a semicubic parabola, when the curvature radius of their curves at the tangency point vanishes at the cusp point. We find the $\A$-normal form of these families, their envelopes and local patterns near the cusp. We obtain a new codimension 2 singularity…
New definition of Bäcklund transformation for surface isometric deformation.
problem Defining Bäcklund transformation in surface isometric deformation.
method Proving generic 4D integrable rolling distribution splits into 1D family of 3D distributions.
result Introducing new definition of Bäcklund transformation.
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…
Paper defines a new invariant for surface immersions.
problem Detecting and classifying jumps in surface immersions.
method Defines an integer-valued function to classify jumps involving quadruple points and triple-line tangencies.
result Classifies quadruple point jumps into five geometrically distinct cases.
Engel CR submanifolds of math.DG
Engel structures on bundles over 3-manifolds in complex 3-space.
problem Embedding bundles over 3-manifolds into complex 3-space with Engel structures.
method Sufficient condition for -bundles to admit immersions/embeddings with complex tangencies defining Engel structures.
result Every oriented -bundle over a closed, oriented 3-manifold admits an immersion with complex tangencies defining Engel structures.
A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements midscribed to the unit sphere centered at the origin, and that these representatives are unique up to Möbius transformations of the sphere. Motivated by this result, various papers investigate the prob…
We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic singularities. We define discrete symplectic invariants - the Lagrangian tangency orders. We use these invariants to distinguish symplectic singularities of classical singularities of planar…
The paper extends Frobenius' Theorem to non-involutive surfaces below threshold.
problem Generalizing Frobenius' Theorem to surfaces with lower regularity.
method Combining fractional Sobolev estimates, Stokes-type theorem, and Lusin's Theorem.
result Sharp exponents for the tangency set's null measure.
We study the singularities of the isotropic skeleton of a Weinstein manifold in relation to Nadler's program of arboreal singularities. By deforming the skeleton via homotopies of the Weinstein structure, we produce a Morse-Bott* representative of the Weinstein homotopy class whose stratified skeleton determines its sy…
Constructs surfaces with specific topologies and curvatures.
problem Creating surfaces with prescribed genus and ends.
method Using a family of constant mean curvature surfaces constructed in \cite{Kleene}, resolving tangency points over catenoidal necks.
result Complete embedded surfaces with freely prescribed genus and ends.
We apply contact homology to obtain new results in the problem of distinguishing immersed plane curves without dangerous self-tangencies.
Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
problem Characterizing domains with Kähler-Einstein Bergman metrics.
method Asymptotics of derivatives of the Bergman kernel along critically tangent paths.
result Two-dimensional pseudoconvex domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
We study the fillability (or embeddability) of 3-dimensional structures under the geometric flows. Suppose we can solve a certain second order equation for the geometric quantity associated to the flow. Then we prove that if the initial structure is fillable, then it keeps having the same property as long as …
Develops Heuristic Portfolio Optimization (HPO) as an information-restricted projection of Markowitz/tangency solution
problem Practitioners allocate capital with forecast-light rules like equal weight, inverse volatility, risk parity, HRP, and RA-HRP
method Implies-return principle and fixed-tree cluster-Sharpe recursion
result Formalizes HPO maps, proves defect equals squared inefficiency, and identifies nodewise alphas as policy-gradient coordinates
We study the following question: given a set P of 3d-2 points and an immersed curve G in the real plane R^2, all in general position, how many real rational plane curves of degree d pass through these points and are tangent to this curve. We count each such curve with a certain sign, and present an explicit formula for…
Researchers classify translators and rotators in hyperbolic 3-space for mean curvature flow.
problem Classifying translators and rotators in hyperbolic 3-space for mean curvature flow.
method Existence and uniqueness proofs, tangency principle application, classification of constant mean curvature translators and rotators.
result Existence and uniqueness of two distinct families of complete rotational translators in hyperbolic 3-space.
Formula conjectured for rational cuspidal curves in projective plane.
problem Counting rational cuspidal curves in projective plane.
method Extending Kontsevich's recursion formula and using geometric input about tangency of curves at nodal points.
result Conjectural formula agrees with earlier computations and extends to rational quartics with E6 singularity.
Study shows critical width for rigidity of equatorial zones on spheres.
problem Mean curvature rigidity of equatorial zones on spheres.
method Used tangency principle and trap-slice lemma for strong rigidity, and constructed nontrivial perturbations using Delaunay surfaces for non-rigidity.
result Critical width exists for rigidity, beyond which zones are non-rigid.
The study analyzes ETFs' portfolio optimization and tail-risk management.
problem Analyzing the performance of actively managed ETFs in managing risk and diversification.
method Daily Bloomberg data for 30 funds, evaluating various strategies under long-only and long-short constraints.
result Tangency-type portfolios generally outperform buy-and-hold benchmarks, while minimum-variance and CVaR-minimizing portfolios sacrifice upside for downside control.
The study proves rigidity and non-rigidity of spherical caps in mean curvature.
problem Understanding mean curvature rigidity and non-rigidity on spherical caps.
method Used a Tangency Principle to prove rigidity and constructed counterexamples for non-rigidity.
result Contrast between rigidity and non-rigidity phenomena on spherical caps.
Consider the standard symplectic $(\RR^{2n}, ω_0)$, a point $p\in\RR^{2n}$ and an immersed closed orientable hypersurface $Σ\subset\RR^{2n}\minus\{p\}$, all in general position. We study the following passage/tangency question: how many lines in $\RR^{2n}$ pass through and tangent to parallel to the 1-dimension…
We generalize Turaev's definition of torsion invariants of pairs , where is a 3-dimensional manifold and is an Euler structure on (a non-singular vector field up to homotopy relative to the boundary of and local modifications in the interior of ). Namely, we allow to have arbitrary boundar…
A new formula detects differences between counterexamples and standard embeddings of circles.
problem Detecting differences between counterexamples and standard embeddings of circles.
method Introducing a Gauss diagram formula for two-component links.
result A desired function is found for the two-component case.
The total curvature of complex hypersurfaces in $\bC^{n+1}$ and its variation in families appear to depend not only on singularities but also on the behaviour in the neighbourhood of infinity. We find the asymptotic loss of total curvature towards infinity and we express the total curvature and the Gauss-Bonnet defect …
The study explores vector flows on manifolds, focusing on polynomial constraints and equivalence relations.
problem Understanding vector flows on compact manifolds with constraints on tangency patterns.
method Introduces equivalence relations (quasitopy) and computes spaces of polynomials with constrained divisors.
result Quasitopy classes stabilize as the degree of polynomials increases.
Portfolio Optimizationq-fin.PM
In this paper Portfolio Optimization techniques were used to determine the most favorable investment portfolio. In particular, stock indices of three companies, namely Microsoft Corporation, Christian Dior Fashion House and Shevron Corporation were evaluated. Using this data the amounts invested in each asset when a po…
In the mid-1980's, M. Gromov used his machinery of the -principle to prove that there exists totally real embeddings of into . Subsequently, Patrick Ahern and Walter Rudin explicitly demonstrated such a totally real embedding. In this paper, we consider the generic situation for such embeddings, …
In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil $ω_λ=ω_2+λω_1=uδ'(x-y)+\f{1}{2}u_xδ(x-y)+λδ'(x-y)$. Deformations are generated by a sequence of vector fields , where each is homogenous of degree with respect to a grading induced by rescali…
We establish a full principle (close, relative, parametric) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respe…
We generalize Turaev's definition of torsion invariants of pairs (M,x), where M is a 3-dimensional manifold and x is an Euler structure on M (a non-singular vector field up to homotopy relative to bM and local modifications in int(M). Namely, we allow M to have arbitrary boundary and x to have simple (convex and/or con…
Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. I…
Study on inflection points of plane curve shadows with fixed embedded shapes.
problem Minimum number of inflection points in plane curves with fixed embedded shadows.
method Finite coorientation problem on building polygons, dynamic programming, universal lower bound, tree-necklace shadows.
result Exact formula for minimum number of normalized inflections for tree-like shadows.
Since the pioneering work of Ghys, Langevin and Walczak among others, it has been known that several methods of dynamical systems theory can be adopted to study of foliations. Our aim in this paper is to investigate complexity of foliations, by generalising existence problem of time averages in dynamical systems theory…
Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R^3, with values in any Abelian group. We show they are all functions of the universal order 1 invariant that we construct as T \oplus P \oplus Q where T is a Z valued invariant reflecting the number of triple poin…
Paper explores relationships between triple chords and a specific homotopy relation in knot theory.
problem Understanding the relationship between triple chords and a homotopy equivalence class in knot theory.
method Analyzes the number of triple chords and their connection to the strong (1, 2) homotopy equivalence class.
result Prime knot projections are trivialized by strong (1, 2) homotopy if they have no triple chords.
Two obstructions found to keep a foliation transverse.
problem Ensuring a foliation remains transverse to a submanifold.
method Two obstructions: determinant-line and twisted homology/degree.
result Obstructions can be used to determine when a foliation is not transverse.
A new asset allocation model uses Markov states from clustered efficient frontier coefficients.
problem Characterizing market regimes using efficient frontiers for better asset allocation.
method Hierarchical clustering of monthly efficient frontier coefficients to define states, then a Markov process on these states for portfolio optimization.
result The model significantly outperforms benchmark portfolios empirically.