Geometric technique determines exactness of SDP robustness certificate.
problem Certifying robustness of neural networks to adversarial examples.
method Geometric projection onto hyperbola, SDP relaxation of ReLU activation.
result SDP certificate is exact for a single hidden layer under mild assumptions.
We introduce canonical coordinates on minimal time-like surfaces in the n-dimensional Minkowski space and prove the existence and the uniqueness of these parameters. With respect to these coordinates the coefficients of the first fundamental form are expressed by the invariants of the surface. On any time-like surface …
Constructs real algebraic maps with specific geometric constraints.
problem Construct smooth functions with prescribed Reeb graphs.
method Explicitly constructs real algebraic maps whose images are domains surrounded by products of hyperbolas and affine spaces.
result New examples of real algebraic maps with specified geometric constraints.
The paper studies a flow of spacelike curves in a Lorentz-Minkowski plane, showing convergence to a constant function.
problem Evolution of spacelike graphic curves in Lorentz-Minkowski plane.
method Anisotropic inverse mean curvature flow with vanishing Neumann boundary condition.
result The evolving curves converge to a constant function as time tends to infinity.
Study local geometry of bi-contact structures on 3-manifolds.
problem Understanding the local geometry of bi-contact structures.
method Investigating maps preserving each contact structure and discovering differential invariants.
result Discovering contact ellipses and hyperbolas, and relating to symplectic structures.
In this paper we prove the existence of families of n-dimensional complete embedded minimal submanifolds of C^n with a prescribed configuration of k>1 asymptotic planes. These submanifolds are obtained by desingularizing the intersection of the asymptotes, using a gluing theorem applied to a generalization of a special…
In this paper we study an experimentally-observed connection between two seemingly unrelated processes, one from computational geometry and the other from differential geometry. The first one (which we call "grid peeling") is the convex-layer decomposition of subsets G⊂Z2 of the integer grid, previous…
In this paper, we present two observations about static spherically symmetric solutions of the Einstein-Klein-Gordon equations. The first is a comment extending the well-known result of the existence of static states (i.e. standing wave solutions) of the Einstein-Klein-Gordon equations. The second more important observ…
The paper explores fully affine maximal curves and their properties.
problem Whether the hyperbola is the fully affine maximal curve in R^2.
method Utilizing evolution equations for curves, the second variational formula for fully affine extremal curves in R^2 was obtained.
result The fully affine maximal curves in R^2 are much more abundant and include explicit curves y=x^α (α is a constant and α∉{0,1,1/2,2}).
We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in 3+1 dimensions as a hypersurface in R4,1. For the Schwarzschild metric the…
We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these curves being centered at a point of a regular curve and contained in its normal p…
We study the second order invariants of a Lorentzian surface in R2,2, and the curvature hyperbolas associated to its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentzian surface and give an extrin…
New method constructs translationally equivariant hyperbolic affine spheres.
problem Constructing translationally equivariant hyperbolic affine spheres.
method Noncompact Iwasawa factorization via DPW method and Weierstrass elliptic functions.
result Every translationally equivariant hyperbolic affine sphere is equiaffinely equivalent to one with a circle, hyperbola, or parabola slice curve.
We construct simply connected, complete, non-CMC biconservative surfaces in the 3-dimensional hyperbolic space H3 in an intrinsic and extrinsic way. We obtain three families of such surfaces, and, for each surface, the set of points where the gradient of the mean curvature function does not vanish is de…
We investigate entropy as a financial risk measure. Entropy explains the equity premium of securities and portfolios in a simpler way and, at the same time, with higher explanatory power than the beta parameter of the capital asset pricing model. For asset pricing we define the continuous entropy as an alternative meas…
The paper studies the convex hull of random points in a triangle, focusing on the asymptotic behavior and phase transitions.
problem Analyzing the convex hull of random points in a triangle with a phase transition.
method Conditional analysis of the convex hull's boundary size and shape, proving phase transitions and convergence to specific curves.
result The convex hull's boundary converges to a hyperbola or parabola under specific conditions, solving an optimization problem.
Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in sub-Lorentzian Heisenberg group.
method First-variation formula derivation and isoperimetric candidates classification.
result Characterization and conjecture of isoperimetric maximizers.
Bi-contact surgery operations can be applied to Anosov flows.
problem Characterizing Anosov flows and their properties.
method Metric and contact geometric characterizations, Liouville geometry, Reeb dynamics.
result Bi-contact surgery operations can be applied to Anosov flows.
Solitons are special polygon midpoints under affine transformations.
problem Characterizing polygons whose midpoints under affine transformations form a new polygon.
method Analyzing midpoints polygons and their relationship to affine transformations and differential equations.
result A large class of polygons are on an orbit of a one-parameter subgroup of the affine group, and these curves are solutions to a specific differential equation.
Study on Hausdorff dimension and curvature bounds in sub-Lorentzian Heisenberg group.
problem Hausdorff dimension and curvature bounds in sub-Lorentzian Heisenberg group.
method Elementary variational approach, Lorentzian isoperimetric problem, uniform estimate of causal diamonds.
result Heisenberg group has Lorentzian Hausdorff dimension 4 and satisfies neither timelike curvature-dimension nor measure contraction properties.
Using the complex parabolic rotations of holomorphic null curves in C4, we transform minimal surfaces in Euclidean space R3⊂R4 to a family of degenerate minimal surfaces in Euclidean space R4. Applying our deformation to holomorphic null curves in ${…
This paper connects billiards in ellipses to focal billiards in ellipsoids.
problem Proving the existence of isometric counterparts between billiards in ellipses and focal billiards in ellipsoids.
method Continuous transition via isometric focal billiards in a fixed ellipsoid.
result Established the connection between planar and spatial billiards.
Paper classifies conic submanifolds in control systems.
problem Characterizing and classifying conic submanifolds in control systems.
method Feedback equivalence of control-affine and fully nonlinear systems.
result Complete description of non-degenerate conic submanifolds.
In a previous paper the author introduced the notion of TreadmillSled of a curve, which is an operator that takes regular curves in R^2 to curves in R^2. This operator turned out to be very useful to describe helicoidal surfaces, for example, it provides an interpretation for the profile curve of helicoidal surfaces wi…
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.
New approach to rotational Weingarten surfaces using geometric momentum.
problem Classifying and characterizing rotational Weingarten surfaces.
method Introducing geometric linear momentum of a plane curve to reduce Weingarten conditions to differential equations.
result Classification of non-degenerate quadric surfaces and elasticoids.
Let L be a Lagrangian submanifold of a pseudo- or para-Kähler manifold which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation of the volume of L with respect to Hamiltonian variations. We apply this formula to several cases. In particular …
A cone projection maps complex plane arcs to conic sections with fixed focus and directrix.
problem Mapping complex plane arcs to conic sections with fixed focus and directrix.
method Elementary spatial construction and reciprocal lens identity.
result The cone projection maps every line not through the origin onto an arc of a conic with focus at the origin and directrix the line itself.
Bayesian inference with deep, weakly nonlinear networks is solved rigorously.
problem Bayesian inference with neural networks of specific structure.
method Perturbative analysis of fully connected neural networks with a shaped nonlinearity.
result Neural network Bayesian inference can be equivalent to kernel methods under certain conditions.
The cone projection fR(z)=z/(1+∣z∣/R) maps lines to conic arcs with specific properties.
problem Mapping lines to conic arcs with specific properties.
method Using a reciprocal lens identity and radial homeomorphism.
result The Self-Directrix Theorem and Confocal--Codirectrix Theorem.
New symmetries discovered in Kepler's orbit family.
problem Symmetry properties of Kepler orbits and related subfamilies.
method Projective geometry and Lie's infinitesimal point symmetries.
result Kepler orbits form a flat family with a 7-dimensional local symmetry group.