arXiv research

In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a…

In this paper we study singular points of the Wigner caustic and affine $λ$--equidistants of planar curves based on shapes of these curves. We generalize the Blaschke-Süss theorem on the existence of antipodal pairs of a convex curve.

Affine $λ$-equidistants of convex polygons with parallel opposite sides have applications to isoperimetric inequalities.

The classical isoperimetric inequality in the Euclidean plane $\mathbb{R}^2$ states that for a simple closed curve $M$ of the length $L_{M}$, enclosing a region of the area $A_{M}$, one gets \begin{align*} L_{M}^2\geqslant 4πA_{M}. \end{align*} In this paper we present the improved isoperimetric inequality, which state…

In this paper we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with non-vanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented areas of the Wigner caustic, the Constant Width Measure Set and the Spherical Meas…

This paper describes caustics of wave fronts reflected by a surface.

We introduce the notions of the caustic-equivalence and the weak caustic-equivalence relations of reticular Lagrangian maps in order to give a generic classification of caustics on a corner. We give the figures of all generic caustics on a corner in a smooth manifold of dimension 2 and 3.

Study of caustics in Einstein-dust system, showing spacetime singularities and diverging curvature.

The secant caustic of a planar curve $M$ is the image of the singular set of the secant map of $M$. We analyse the geometrical properties of the secant caustic of a planar curve, i.e. the number of branches of the secant caustic, the parity of the number of cusps and the number of inflexion points in each branch of thi…

Study caustics of an elliptical paraboloid and extend Apollonius problem solution.

Researchers found the Wigner derivative and its inverse are equal for spherical tetrahedra.

Optical interpretation of Euler's angle problem for caustics of light rays.

The paper shows that oval caustics have at least 4 cusps.

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…

Bertrand framed surfaces defined in Euclidean 3-space with applications.

In the Minkowski space-time, a world hyper-sheet is a timelike hypersurface consisting of a one-parameter family of spacelike submanifolds. Recently, Bousso and Randall introduced the notion of caustics of world hyper-sheets in order to define the notion of holographic domains in space-time. Here, we give a mathematica…

We provide a geometric explanation for the existence of magnification relations for the A, D, E family of caustic singularities, which were established in recent work. In particular, it was shown that for families of general mappings between planes exhibiting any of these caustic singularities, and for any non-caustic …

The paper shows caustics by reflection in projective Finsler metrics have at least four cusps.

Unified view of surfaces in R^n using Gauss map, caustics, and quadratic forms.

The paper studies the number of normals to ellipsoids and their intersections with caustics.

We study first order local invariants of Vassiliev type for Lagrangian immersions with generic planar caustics. For this we produce some examples of 2-parameter families of Lagrangian maps and study their bifurcation diagrams.

We investigate genericities of reticular Lagrangian maps and reticular Legendrian maps in order to give generic classifications of caustics and wavefronts generated by a hypersurface germ without or with a boundary in a smooth manifold.

After recalling the notion of caustics of plane curves and basic equations, we first show the birationality of the caustic map for a general source point S in the plane. Then we prove more generally a theorem for curves D in the projective space of 3x3 symmetric matrices B. For a general 3x1 vector S the projection to …

We show that, for each alpha in the interval (-1,1), the only Riemannian metrics on the space of positive definite matrices for which the alpha and -alpha-connections are mutually dual are matrix multiples fo the Wigner-Yanase-Dyson metric. If we further impose that the metric be monotone, then this set is reduced to s…

A world sheet in anti-de Sitter space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds. We consider the family of lightlike hypersurfaces along spacelike submanifolds in the world sheet. The locus of the singularities of lightlike hypersurfaces along spacelike submanifolds forms …

Wigner's theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the Hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalisation of Wigner's theorem: a holomorph…

We study the equilibrium positions of three points on a convex curve under influence of the Coulomb potential. We identify these positions as orthotripods, three points on the curve having concurrent normals. This relates the equilibrium positions to the caustic (evolute) of the curve. The concurrent normals can only m…

Optimal spectral method found for inhomogeneous spiked Wigner model.

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}…

After Galvez, Martinez and Milan discovered a (Weierstrass-type) holomorphic representation formula for flat surfaces in hyperbolic 3-space, the first, third and fourth authors here gave a framework for complete flat fronts with singularities in H^3. In the present work we broaden the notion of completeness to weak com…

The paper proves the conjecture about the number of cusps in a caustic formed by reflecting rays in a circle.

In this paper we introduce the Constant Width Measure Set, which measures the constant width property of an oval, i.e. the planar simple closed strictly convex curve. We study its geometrical properties. We find the exact relation between the length and the area of the region bounded by an oval $M$. Namely, the followi…

Study on eigenvalue distribution of correlated time series deforming the semi-circle law.

Extends Wigner's representation to study super hyperbolic geometry.

A novel method computes Wigner kernels for atomic environments, achieving state-of-the-art accuracy.

The study improves norms of spectral projectors on specific surfaces.

Consistent model selection for spiked Wigner model via AIC-type criteria.

Study detects signals in spiked Wigner models using log likelihood ratio.

We describe the global structure of holomorphic webs in codimension one, and in particular their singularity (caustic). Various concepts are introduced, which have no interest locally near a regular point, such as the type, the reducibility, the quasi-smoothness, the CI property (complete intersection), the dicriticity…

Billiard motion in ellipses analyzed with canonical coordinates.

Study describes singularities of distance squared functions on singular surfaces.

On the manifold of positive definite matrices, we investigate the existence of pairs of flat affine connections, dual with respect to a given monotone metric. The connections are defined either using the $α$-embeddings and finding the duals with respect to the metric, or by means of contrast functionals. We show that i…

Extends particle classification to curved space-times using groupoids.

Study of correlated Wigner matrices with BBP transitions.

Paper develops new method for detecting latent structure in large symmetric data matrices.

We consider the weak detection problem in a rank-one spiked Wigner data matrix where the signal-to-noise ratio is small so that reliable detection is impossible. We propose a hypothesis test on the presence of the signal by utilizing the linear spectral statistics of the data matrix. The test is data-driven and does no…

We present an original and novel method based on random matrix approach that enables to distinguish the respective role of temporal autocorrelations inside given time series and cross correlations between various time series. The proposed algorithm is based on properties of Wigner eigenspectrum of random matrices inste…

Study asymptotics of unitary matrix elements in quantum mechanics.