arXiv research

New Lie group approach for envelope surface computation.

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.

New method for operating envelope identifies key performance indicators without arbitrary binning.

Dehn quandles of groups and surfaces unify various quandle constructions.

In the present paper, we introduce the new construction of quandles. For a group $G$ and its subset $A$ we construct a quandle $Q(G,A)$ which is called the $(G,A)$-quandle and study properties of this quandle. In particular, we prove that if $Q$ is a quandle such that the natural map $Q\to G_Q$ from $Q$ to its envelopi…

The Galilean group is the group of symmetries of Newtonian mechanics, with Lie a lgebra $\gal(n)$. We find algebraically independent generators for the center of the universal enveloping algebra of $\gal(n)$ using coadjoint orbits.

Correct method found for drawing precise envelope of straight lines.

pFedMe uses Moreau envelopes to improve personalized FL performance.

Study three discrete envelope types of polygon bisection lines.

Paper solves four problems of pseudo-circle envelopes in Minkowski plane.

Describes envelopes of Thurston metric on Teichmüller space.

Improves regression efficiency by separating material and immaterial parts of responses.

Study circle families' envelopes and related curves.

We construct an action of the braid group B_N on the twisted quantized enveloping algebra U'_q(o_N) where the elements of B_N act as automorphisms. In the classical limit q -> 1 we recover the action of B_N on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action prese…

Solves four problems related to circle families in the plane.

Solves four problems related to sphere families in 3D space.

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.

Novel approach reduces DNN redundancy and enhances computational efficiency.

The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (i.e., the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-di…

Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo…

MoreauGrad interprets neural nets robustly and sparsely.

We establish a canonical correspondence between connected quandles and certain configurations in transitive groups, called quandle envelopes. This correspondence allows us to efficiently enumerate connected quandles of small orders, and present new proofs concerning connected quandles of order p and 2p. We also present…

Extends orbital integral evaluation to center of enveloping algebra.

Study on Laplacian determinant in isosceles triangles, finding equilateral triangle minimizes determinant.

Enhances neural network robustness with polyhedral envelope regularization.

Separation of the sources and analysis of their connectivity have been an important topic in EEG/MEG analysis. To solve this problem in an automatic manner, we propose a two-layer model, in which the sources are conditionally uncorrelated from each other, but not independent; the dependence is caused by the causality i…

This work uses Lasry-Lions envelopes to solve nonconvex optimization problems.

We make a systematic study of (quasi-)plurisubharmonic envelopes on compact Kähler manifolds, as well as on domains of $\mathbb{C}^n$, 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…

New geometric mechanism solves four envelope problems.

New methods help escape strict saddle points in nonsmooth optimization.

This paper generalizes the envelope of mid-lines to intermediate lines for a plane curve.

The paper proves boundedness of envelopes in complex manifolds.

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 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…

Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide examples of G-categories.

Smooth convergence to an enveloping cylinder proved for mean curvature flow of complete graphical hypersurfaces.

We study the geometry of Outer Space $CV_n$ in regard of the asymmetric Lipschitz metric via envelopes, that is the set of all geodesics between two points. In the simplicial structure of $CV_n$ the envelopes are polytopes. We construct a piecewise unique geodesic between any two points in $CV_n$ by concatenating edges…

Defines a calculus for integrating Moreau envelopes in differentiable programming.

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…

Given any smooth plane curve α(s)representing a mirror that reflects light the usual way and any radiant light source at a point in the plane, the reflected light will produce a caustic envelope. For such an envelope, we show that there is an associated curve \b{eta}(s) and a family of circles C(s) that roll on \b{eta}…

A spacetime can be embedded in an enveloping space with all its extensions.

Let G be a complex semisimple Lie group, K a maximal compact subgroup and V an irreducible representation of K. Denote by M the unique closed orbit of G in P(V) and by O its image via the moment map. For any measure on M we construct a map from the Satake compactification of G/K (associated to V) to the Lie algebra of …

The envelope of straight lines affine normal to a plane curve C is its affine evolute; the envelope of the affine lines tangent to C is the original curve, together with the entire affine tangent line at each inflexion of C. In this paper, we consider plane curves without inflexions. We use some techniques of singulari…

The paper extends a theorem to Lie-Rinehart algebras and provides new decompositions of universal enveloping algebras.

The group of real 4 by 4 upper triangular matrices with 1s on the diagonal has a left-invariant subRiemannian (or Carnot-Caratheodory) structure whose underlying distribution corresponds to the superdiagonal. We prove that the associated subRiemannian geodesic flow is not completely integrable. This provides the first …

In this paper, we further study the forward-backward envelope first introduced in [28] and [30] for problems whose objective is the sum of a proper closed convex function and a twice continuously differentiable possibly nonconvex function with Lipschitz continuous gradient. We derive sufficient conditions on the origin…