Euler and Delisle developed a map method for the Russian Empire, which is now outperformed by the Lambert conformal conical projection.
problem Mapping a country onto a flat map while minimizing distortion.
method Developed a heuristic method for mapping the Russian Empire, which was later named Delisle--Euler map.
result The Lambert conformal conical projection outperforms the Delisle--Euler map in several respects.
This research examines anisotropic conformal transformations of pseudo-Finsler surfaces.
problem Investigating anisotropic conformal transformations of conic pseudo-Finsler surfaces.
method Using modified Berwald frame, the study finds necessary and sufficient conditions for anisotropic conformal transformations and analyzes geometric properties.
result Necessary and sufficient conditions for anisotropic conformal transformations of pseudo-Finsler surfaces are derived.
Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.
problem Finding conditions for conformal deformations to constant scalar curvature in conic metrics.
method Analyzes conformal deformations within a class of incomplete Riemannian metrics that generalize conic orbifold singularities.
result Determines sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1.
The abstract discusses a new causal structure on manifolds using paths and points.
problem Constructing a causal structure on manifolds using paths and points.
method Constructing a four-manifold from pairs of points and paths, and a seven-dimensional manifold from pairs of points and conics.
result The causal structure corresponds to a conformal structure only when the underlying surface is a real projective plane.
We describe the range of the Radon transform on the space M of irreducible conics in $\CP^2$ in terms of natural differential operators associated to the SO(3)-structure on M=SL(3,R)/SO(3) and its complexification. Following \cite{moraru} we show that for any function F in this range, the zero locus of F is…
Solves probabilistic Lambert problem connecting astrodynamics with optimal mass transport.
problem Determining spacecraft velocity for given positions with probabilistic constraints.
method Generalized optimal mass transport (OMT) and Schrödinger bridge (SBP) connections.
result Existence and uniqueness of solution for probabilistic Lambert problem.
Proves positive mass theorem on conical manifolds with small angles.
problem Proving the positive mass theorem on conical manifolds with small cone angles.
method Analyzes conical manifolds with small cone angles, assuming spin structure and locally conformal flatness.
result Proves the positive mass theorem under specified conditions.
Constructs metrics on Riemann surfaces with singularities.
problem Creating constant curvature metrics on surfaces with specific singularities.
method Using meromorphic 1-forms and ODEs to construct conformal metrics.
result Classified constant curvature metrics on S2 with two conical singularities. The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.
problem Understanding non-differentiability points in limit sets of convex projective structures.
method Introduces hyperplane conicality for θ-Anosov representations and uses it to prove properties of boundary maps. result Hilbert entropy is linked to the Hausdorff dimension of non-differentiability points in flag spaces.
Improved pricing method for illiquid assets using Lambert function.
problem Inaccurate pricing of illiquid assets using traditional methods.
method Deterministic decomposition of reservation price using Lambert function; improved Monte Carlo method (LMC).
result Improved accuracy in pricing illiquid assets through LMC method.
Proves mass theorem for AF manifolds with conical singularities.
problem Proving the positive mass theorem for specific types of manifolds.
method Conformal blow up technique applied to AF manifolds with isolated conical singularities.
result Positive mass theorem proven for the specified manifolds.
Survey on metrics with conic singularities on Riemann surfaces.
problem Conformal metrics with conic singularities on Riemann surfaces.
method Study of metrics with constant positive curvature and conic singularities.
result Simplified proofs for interesting cases, including those by C. L. Chai, C. S Lin, and C. L. Wang.
lamBERT learns language and actions using multimodal BERT.
problem Learning language and actions in complex environments.
method Extending BERT to multimodal representation and integrating with reinforcement learning.
result lamBERT model achieved higher rewards in multitask and transfer settings.
Delisle's projection explained by Euler in 18th century.
problem Geographical projection issues.
method Analyzing Euler's work on Delisle's projection.
result Important mathematical points on metric geometry of surfaces.
In this paper, we establish a framework for the analysis of linear parabolic equations on conical surfaces and use them to study the conical Ricci flow. In particular, we prove the long time existence of the conical Ricci flow for general cone angle and show that this solution has the optimal regularity, namely, the ti…
Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.
problem Existence of conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundaries.
method Variational argument to derive existence results for surfaces with at least two boundary components.
result First result in this setting for surfaces with conical singularities of both positive and negative orders.
Study on conical singularities in 2D surfaces, deriving Polyakov formulas.
problem Analyzing zeta-regularized determinants in surfaces with conical singularities.
method Demonstrated variational and integrated Polyakov formulas for conical singularities, circular sectors, and cones.
result Explicit formulas for the determinant of conical sectors and cones derived.
The paper proves geodesics and conic sections are length-minimizing under specific metrics.
problem Finding shortest paths in complex geometries.
method Calibrations and conformal metrics.
result Geodesics and conic sections are length-minimizing.
We present an algorithm for converting an indoor spherical panorama into a photograph with a simulated overhead view. The resulting image will have an extremely wide field of view covering up to 4π steradians of the spherical panorama. We argue that our method complements the stereographic projection commonly used in t…
Characterizes representations for complex projective structures with specific branch data.
problem Understanding representations of surface groups as holonomy of complex projective structures.
method Computing holonomies for spherical metrics and affine structures with prescribed conical angles.
result Computed holonomies for spherical metrics and affine structures with specific conical angles.
Researchers calculated EVaR for various distributions using Lambert function.
problem Difficulty in finding analytical representation of EVaR measure.
method Used Lambert function to calculate EVaR for multiple distributions.
result Successfully calculated EVaR for 7 specific distributions.
We consider the constant Q-curvature metric problem in the given conformal class on conic 4-manifolds and study related differential equations.
The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.
problem Proving Strichartz and spectral projection theorems on curved surfaces.
method Using large negative curvature neighborhoods, the study proves theorems on asymptotically conic and Euclidean ends surfaces.
result The study proves theorems without loss of interval on specific types of curved surfaces.
Study on special anisotropic conformal changes of conic pseudo-Finsler surfaces.
problem Exploring various anisotropic conformal transformations of conic pseudo-Finsler surfaces.
method Presented various anisotropic conformal transformations including C-anisotropic, horizontal C-anisotropic, and vertical C-anisotropic transformations. result Vertical φT-condition transformation makes every Landsberg surface Berwaldian. The goal of this article is to generalise the Witten deformation to even dimensional conic manifolds and a class of functions called admissible Morse functions.
The paper proves a Penrose inequality for 4-discs with hyperbolic ends and conic singularities.
problem Proving a Penrose inequality for conformal metrics on a unit 4-disc with hyperbolic ends and conic singularities.
method Defining a mass term and proving a Penrose type inequality with curvature condition.
result Proves a Penrose inequality for conformal metrics on a unit 4-disc with hyperbolic ends and conic singularities.
Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.
problem Existence and uniqueness of conformal metrics with negative curvature and singularities.
method Proves existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
result Existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
Infinite circle packings on surfaces with conical singularities are possible.
problem Finding hyperbolic metrics with prescribed angles and circle packings on surfaces with punctures.
method Using infinite triangulations and hyperbolic metrics, the approach involves identifying the underlying Riemann surface and ensuring the circle packing combinatorics match the given triangulation.
result There are infinitely many conical hyperbolic structures in a conformal class with a circle packing in the combinatorics of a given triangulation.
We study the asymptotics of the determinant of Laplacian on a translation surface (a compact Riemann surface equipped with a conformal flat conical metric with trivial holonomy) of genus g with 2g-2 conical points of angle 4πas two conical points collide.
A necessary and sufficient condition for the existence and uniqueness of a conformal metric on 2-sphere of constant curvature 1 and with three conical singularities of prescribed order is given.
A conformal metric g with constant curvature one and finite conical singularities on a compact Riemann surface Σ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function f on Σ\{singularities}, called the {\it developing …
Study on prescribing positive curvature with conical singularities on a sphere.
problem Prescribing positive curvature with conical singularities on a sphere.
method Fine analysis of bubble trees and an area identity in the convergence process.
result Criterion for nonexistence in an open region of the prescribing data.
Explains mapping properties of elliptic operators in conical spaces.
problem Understanding mapping properties of geometric elliptic operators in conical spaces.
method Develops an approach based on B.-W. Schulze's work.
result Illustrates versatility of results in Geometric Analysis.
We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature 1 having a single conical singularity of angle 4π.
The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.
problem Analyzing conical Kähler-Einstein metrics on rank one horosymmetric Fano manifolds.
method Investigates the convergence of these metrics as the cone angle approaches zero, focusing on the behavior on the complement of a codimension one orbit.
result The metrics converge to the Kähler-Einstein metric on the basis and Stenzel's Ricci flat Kähler metrics on the fibers.
New approach to prescribing Gaussian curvature on spheres with conical singularities.
problem Prescribing Gaussian curvature on the 2-sphere with conical singularities.
method Variational methods not relying on Moser-Trudinger inequality, plus precompactness theorem.
result Sufficient conditions for a positive function to be the Gaussian curvature of a conformal conical metric.
The "dancing metric" is a pseudo-riemannian metric g of signature (2,2) on the space M4 of non-incident point-line pairs in the real projective plane RP2. The null-curves of (M4,g) are given by the "dancing condition": the point is moving towards a point on the line, about which the li…
Proves existence of Yamabe metrics on conical 4-manifolds using min-max method.
problem Existence of Yamabe metrics on conical 4-manifolds with singular points.
method Min-max scheme adapted to singular setting, leveraging recent positive mass theorems.
result Existence of Yamabe metrics on conical 4-manifolds with finitely-many singular points.
We study a behavior of the conformal Laplacian operator Łg on a manifold with \emph{tame conical singularities}: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the operator Łg on such manifolds. We describe the asymptotic of a general solution …
Geometrically constructs twist-field correlation functions in CFT.
problem Understanding entanglement entropy in quantum systems.
method Using Cauchy-Hadamard renormalization of Polyakov anomaly integral on surfaces with conical singularities.
result Provides a purely mathematical interpretation of entanglement entropy results.
Study curve shortening flow on Riemann surfaces with conical singularities.
problem Evolution of curves on Riemann surfaces with singular points.
method Curve shortening flow governed by a degenerate quasilinear parabolic equation.
result Evolving curves stay fixed at singular points and show collapsing and convergence results.
We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization the…
In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields …
Derives formulas for determinant of Laplacian on curved surfaces.
problem Calculating the determinant of the Laplacian on higher genus polyhedral surfaces.
method Variational formulas derived with respect to conical points and angles.
result Explicit expression for determinant up to moduli-dependent factor.
An HCMU metric is a conformal metric which has a finite number of singularities on a compact Riemann surface and satisfies the equation of the extremal Kähler metric. In this paper, we give a necessary and sufficient condition for the existence of a kind of HCMU metrics which has both cusp singularities and conical sin…
The paper generalizes a mean value theorem for solutions of the ultrahyperbolic equation.
problem Establishing a mean value property for solutions of the ultrahyperbolic equation.
method Using conformal maps of the pseudo-Euclidean space of signature 2+2, the paper extends Asgeirsson's theorem to a more general class of pairs of curves.
result The mean value property is proven for non-degenerate conjugate conics, including conjugate circles, hyperbolae, parabolae, and line-empty pairs.
The virtual dimensions of both framed and unframed SU(2) magnetic monopoles on asymptotically conic 3-manifolds are obtained by computing the index of a Fredholm extension of the associated deformation complex. The unframed dimension coincides with the one obtained by Braam for conformally compact 3-manifolds. The comp…
The paper solves navigation problems on conic Kropina manifolds and establishes curvature relationships.
problem Navigation problems on conic Kropina manifolds.
method Analyzes the solution of navigation problems and establishes curvature relationships.
result The solution to navigation problems on conic Kropina manifolds must be either a Randers metric or a Kropina metric.