The paper proves boundedness of envelopes in complex manifolds.
problem Regularity of envelopes in complex manifolds.
method Analyzes bounded functions and their envelopes in the context of cohomology classes and Laplacians.
result The α-psh envelope P(f) is locally bounded with locally bounded Laplacian on the ample locus of {α}. We make a systematic study of (quasi-)plurisubharmonic envelopes on compact Kähler manifolds, as well as on domains of Cn, by using and extending an approximation process due to Berman [Ber13]. We show that the quasi-psh envelope of a viscosity super-solution is a pluripotential super-solution of a given co…
We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.
Study of complex Hessian equations using subharmonic functions and geodesics.
problem Understanding geodesics within complex Hessian equations.
method Perron envelope construction, comparison principle, rooftop equality, Kiselman minimum principle.
result Established criterion for geodesic connectivity among m-subharmonic functions. We study the envelopes of meromorphy of neighborhoods of symplectically immersed two-spheres in complex Kähler surfaces using the Gromov's theory of pseudoholomorphic curves. The construction of a complete family of holomorphic deformations of a non-compact complex curve in a complex manifold, parametrized by a finite …
Study confirms geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.
problem Confirming geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.
method Examined geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains.
result Affirmative answer to a longstanding open question about geodesic connectivity and rooftop envelopes.
We establish plurisubharmonicity of the envelope of Poisson and Lelong functionals on almost complex manifolds. That is, we generalize the corresponding results for complex manifolds and almost complex manifolds of complex dimension two. We also provide some applications to the regularization of J-plurisubharmonic func…
The goal of this work is to prove the regularity of certain quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes …
Study small eigenvalues of Toeplitz operators and their relation to Mabuchi geodesics.
problem Analyzing small eigenvalues of Toeplitz operators on complex projective manifolds.
method Proving the existence of exponentially decaying eigenvalues for Toeplitz operators with specific symbols, and establishing a connection to Mabuchi geodesics.
result Logarithmic distribution of small eigenvalues correlates with Mabuchi geodesics between polarizations.
The paper generalizes envelope constructions for chords in circles, revealing complex singularities.
problem Understanding envelopes of chords in circles with varying parameters and configurations.
method Generalizing the embroidery method to rational and concentric circles, breaking symmetry to reveal higher singularities.
result Higher singularities like swallowtails and butterflies can be unfolded, revealing their structure.
We study holomorphic extensions of Matsuki orbits in complex Grassmannians.
problem Analyzing the analytic continuation of Matsuki orbits in complex Grassmannians.
method Using Rossi's theory of holomorphic extension and the holomorphic fiber bundle structure, we establish that the envelope of holomorphy of each Matsuki orbit coincides biholomorphically with the containing K-orbit. result The envelope of holomorphy of each Matsuki orbit coincides biholomorphically with the containing K-orbit. Develops parabolic pluripotential theory for complex flows.
problem Complex Monge-Ampère equations in degenerate settings.
method Study of semi-concave envelopes and unique solutions.
result Shows semi-concave envelopes as unique solutions.
New framework reduces minimax regret for high-dimensional data.
problem Minimizing regret in high-dimensional data with logarithmic loss.
method Developed envelope complexity framework and spike-and-tails prior.
result Achieves minimax regret within a factor of two over high-dimensional ℓ1-balls. New method for complex Monge-Ampère equations on Kähler manifolds.
problem Degenerate complex Monge-Ampère equations on complex manifolds.
method New approach relying on compactness and envelopes properties of quasi-plurisubharmonic functions.
result New and efficient proofs of fundamental results in Kähler geometry.
In this paper we aim to understand the category of stable-Yetter-Drinfeld modules over enveloping algebra of Lie algebras. To do so, we need to define such modules over Lie algebras. These two categories are shown to be isomorphic. A mixed complex is defined for a given Lie algebra and a stable-Yetter-Drinfeld module o…
Correct method found for drawing precise envelope of straight lines.
problem Widespread method fails to represent the precise shape of envelope.
method Recently discovered correct method for straight line families in the plane.
result Correct method precisely represents the envelope of straight lines.
A nonparametric method for time series analysis extracts envelopes, detects peaks, and clusters data.
problem Extracting envelopes, detecting peaks, and clustering in time series data.
method Iterative procedure that minimizes L1 drift to create upper and lower bounding signals, using Viterbi-like path tracking and optimal elimination rules. result Efficiently calculated solution with near-linear time complexities for various applications.
Determines algebra structure of complex differential forms operators.
problem Identifying the algebra structure of differential operators on complex-valued differential forms.
method Shows it is the universal enveloping algebra of a graded Lie algebra and determines its cohomology.
result Determines the cohomology of the graded Lie algebra with respect to various inner differentials.
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.
Paper solves four problems of pseudo-circle envelopes in Minkowski plane.
problem Four problems of pseudo-circle envelopes in Minkowski plane.
method Solutions to four basic problems.
result Solved four problems of pseudo-circle envelopes in Minkowski plane.
Describes envelopes of Thurston metric on Teichmüller space.
problem Characterizing the shape and properties of envelopes in Teichmüller space.
method Using harmonic stretch lines and topological invariants, the shape and properties of envelopes are described.
result Envelopes are contractible and vary continuously with endpoints.
Let X be a compact complex manifold equipped with a smooth (but not necessarily positive) closed form theta of one-one type. By a well-known envelope construction this data determines a canonical theta-psh function u which is not two times differentiable, in general. We introduce a family of regularizations of u, param…
Improves regression efficiency by separating material and immaterial parts of responses.
problem Improving estimation efficiency in nonlinear multivariate regressions.
method Kernel envelope (KENV) estimator for nonparametric response envelopes in reproducing kernel Hilbert space.
result KENV achieves lower in-sample prediction risk than kernel ridge regression in non-trivial immaterial components.
Study circle families' envelopes and related curves.
problem Understanding relationships between circle families and special curves.
method Investigate envelopes of circle families and their connections to evolutes, pedals, evolutoids, and pedaloids.
result Characterized relationships between circle families and related curves.
New method for operating envelope identifies key performance indicators without arbitrary binning.
problem Accurate identification of operating envelope for optimal KPIs.
method Regularized GA algorithm with interpretability and implementability constraints.
result Validated through simulations and real-world application in mining.
Solves four problems related to circle families in the plane.
problem Four basic problems of circle families in the plane.
method Solves all four basic problems of circle families in the plane.
result All four basic problems are solved.
New Lie group approach for envelope surface computation.
problem Efficient computation of envelope surfaces.
method Interpreting surfaces as curves in Lie group spaces, leveraging Lie group and algebra formalisms.
result Explicit rational parameterization of cone envelope surfaces and solution to trimming problem.
Solves four problems related to sphere families in 3D space.
problem Four basic problems of sphere families in Euclidean 3-space.
method Solves all four basic problems of sphere families in Euclidean 3-space.
result All four basic problems are solved.
The dynamical analysis of American options has motivated the development of robust versions of the classical Snell envelopes. The cost of superhedging an American option is characterized by the upper Snell envelope. The infimum of the arbitrage free prices is characterized by the lower Snell envelope. In this paper we …
This paper explores geometric insights into discrete R-congruences and their envelopes.
problem Understanding the ambiguity in discrete R-congruences and their envelopes.
method Analyzes discrete R-congruences that are enveloped by specific types of surfaces and maps.
result Discovers a 2-parameter family of discrete enveloping surfaces for discrete R-congruences.
New method solves complex Monge-Ampère equations on hermitian manifolds.
problem Solving degenerate complex Monge-Ampère equations on hermitian manifolds.
method New approach using compactness and envelopes properties of quasi-plurisubharmonic functions.
result New relative a priori estimates and existence results for degenerate complex Monge-Ampère equations.
Extends orbital integral evaluation to center of enveloping algebra.
problem Evaluate semisimple orbital integrals for arbitrary elements in the center of the enveloping algebra.
method Explicit geometric evaluation of Casimir operator to arbitrary elements in the center of the enveloping algebra.
result Extension of orbital integral evaluation to center of enveloping algebra.
Study on Laplacian determinant in isosceles triangles, finding equilateral triangle minimizes determinant.
problem Finding the minimum of the spectral determinant on isosceles triangles.
method Analyzing the determinant of the Laplacian on Euclidean isosceles triangle envelopes of fixed area.
result Equilateral triangle envelope minimizes the determinant of the Laplacian.
Enhances neural network robustness with polyhedral envelope regularization.
problem Improving neural network robustness against adversarial attacks.
method Introduces polyhedral envelope regularization to bound the robustness region.
result Demonstrates improved robustness guarantees with minimal computational overhead.
This work uses Lasry-Lions envelopes to solve nonconvex optimization problems.
problem Nonconvex and nonsmooth terms in optimization problems.
method Develops a homotopy approach using Lasry-Lions envelopes to approximate and solve the original problem.
result The method can solve composite minimization problems and is more effective than classical alternatives in certain domains.
Geometric study of Outer Space using envelopes and geodesics.
problem Understanding the geometry and structure of Outer Space CVn. method Study of envelopes in the asymmetric Lipschitz metric of CVn. result For almost all pairs of points in CVn, their envelopes have dimension 3n−4. We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models of the associated geometric notions and objects. Thus, we introduce discrete ana…
Study bounds on Monge-Ampère volumes for degenerate complex equations.
problem Bounds on volumes of Monge-Ampère measures for degenerate complex equations.
method Fine use of quasi-plurisubharmonic envelopes.
result Established a transcendental version of the Grauert-Riemenschneider conjecture.
This is a survey of some of the recent developments in the theory of complex Monge-Ampere equations. The topics discussed include refinements and simplifications of classical a priori estimates, methods from pluripotential theory, variational methods for big cohomology classes, semiclassical constructions of solutions …
New geometric mechanism solves four envelope problems.
problem Four basic problems on envelopes created by hyperplane families.
method Simple geometric mechanism of intersections of perpendicular bisectors and normal lines.
result Solves all four basic problems on envelopes at once.
Our aim is to introduce and advocate non-Σ (non-symmetric) modular operads. While ordinary modular operads were inspired by the structure of the moduli space of stable complex curves, non-Σ modular operads model surfaces with open strings outputs. An immediate application of our theory is a short proof that the mod…
New methods help escape strict saddle points in nonsmooth optimization.
problem Escaping strict saddle points in nonsmooth optimization.
method An inexact stochastically perturbed gradient method applied to the Moreau envelope.
result A variety of algorithms for nonsmooth optimization can efficiently escape strict saddle points of the Moreau envelope.
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. We prove that the forgetful functor from groupoids to pregroupoids has a left adjoint, with the front adjunction injective. Thus we get an enveloping groupoid for any pregroupoid. We prove that the category of torsors is equivalent to that of pregroupoids. Hence we also get enveloping groupoids for torsors, and for pri…
We give a global description of envelopes of geodesic tangents of regular curves in (not necessarily convex) Riemannian surfaces. We prove that such an envelope is the union of the curve itself, its inflectional geodesics and its tangential caustics (formed by the conjugate points to those of the initial curve along th…
We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss ℓ can control the test error under all Moreau envelopes of the loss ℓ. We use our finite-sample bound to directly recover…
By using the support function on the xy-plane, we show the necessary and sufficient conditions for the existence of envelopes of horizontal lines in the 3D-Heisenberg group. A method to construct horizontal envelopes from the given ones is also derived, and we classify the solutions satisfying the construction.
We view conformal surfaces in the 4--sphere as quaternionic holomorphic curves in quaternionic projective space. By constructing enveloping and osculating curves, we obtain new holomorphic curves in quaternionic projective space and thus new conformal surfaces. Applying these constructions to Willmore surfaces, we show…