arXiv research

Study on Jones polynomials and their roots in the unit circle and complex plane.

We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in $\R^2$. Finally, we give a list…

We show that time-dependent fluctuations $\{Δx\}$ in foreign exchange rates are accurately described by a random walk in a complex plane that is demarcated into the gain (+) and loss (-) sectors. $\{Δx\}$ is the outcome of $N$ random steps from the origin and $|Δx|$ is the square of the Euclidean distance of the final …

It is showed that on a plane with a radial density the Four Vertex Theorem holds for the class of all simple closed curves if and only if the density is constant. But for the class of simple closed curves that are invariant under a rotation about the origin, the Four Vertex Theorem holds for every radial density.

New isoperimetric inequalities in the plane with radial weights identified.

We survey the construction and properties of the Yamada polynomial of spatial graphs and present the Yamada polynomial formulae for some classes of graphs. Then we construct an infinite family of spatial graphs for which roots of Yamada polynomials are dense in the complex plane.

In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in $\Bbb R^2.$ The classification gives some interesting phenomena and consequences including: the family of curves conv…

Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.

We present the theory of pseudodifferential operators acting on a vector orbibundle over an orbifold, construct the zeta function of an elliptic pseudodifferential operator and show the existence of a meromorphic extension to the complex plane with at most simple poles. We give formulas for generalized densities on the…

Study on evolving interfaces with complex curvature and density effects.

We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$-manifold. Our estimates are polynomials in the tight areas and Bowen-Margulis-Sullivan densities of geodesic planes, with degree given by the modified critical expo…

We study surfaces in Euclidean space ${\mathbb R}^3$ that are minimal for a log-linear density $φ(x,y,z)=αx+βy+γy$, where $α,β,γ$ are real numbers not all zero. We prove that if a surface is $φ$-minimal foliated by circles in parallel planes, then these planes are orthogonal to the vector $(α,β,γ)$ and the surface must…

A non-Hermitean extension of paradigmatic Wishart random matrices is introduced to set up a theoretical framework for statistical analysis of (real, complex and real quaternion) stochastic time series representing two "remote" complex systems. The first paper in a series provides a detailed spectral theory of non-Hermi…

The study examines stationary integral varifolds near multiplicity 2 planes, proving regularity under specific conditions.

In $\mathbb{R}^n$ with a density $e^ψ$, we study the mean curvature flow associated to the density ($ψ$-mean curvature flow or $ψ$MCF) of a hypersurface. The main results concern with the description of the evolution under $ψ$MCF of a closed embedded curve in the plane with a radial density, and with a statement of sub…

Study vortex loops as coadjoint orbits of diffeomorphisms.

Proves regularity for Brakke flow near stationary half-plane.

Study local and global aspects of complex plane curve embeddings.

In this article we study the tangent cones at first time singularity of a Lagrangian mean curvature flow. If the initial compact submanifold is Lagrangian and almost calibrated by ReΩin a Calabi-Yau n-fold (M,Ω), and T>0 is the first blow-up time of the mean curvature flow, then the tangent cone of the mean curvature f…

Stable planes are locally isomorphic to classical projective planes.

The paper uses complex-valued functions to simplify plane differential geometry and kinematics.

We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank $1$ cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property was introduced in the recent work of McMullen, Mohammadi and Oh.…

We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.

Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.

A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes.

Study shows no smooth embeddings of rational homology balls into complex projective plane.

Lecture notes on curves in complex projective plane from a topological viewpoint.

We propose the study of a conformally invariant functional for surfaces of complex projective plane which is closely related to the classical Willmore functional. We show that minimal surfaces of complex projective plane are critical for this functional and construct some minima for it via the twistors spaces of comple…

Outer billiards studied in complex hyperbolic plane, proving smooth and symplectic properties.

Classifies degenerations of complex projective plane with rational singularities.

New method improves sampling from high-dimensional target densities.

We prove a relative isoperimetric inequalities for Lagrangian half disks in $\mathbb{C}^2$ with respect to a Lagrangian plane, or a complex plane, or a union of any two of Lagrangian or complex planes that intersect transversally at the origin.

Two new proofs classify complete totally geodesic subsets of complex hyperbolic plane.

We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as random, real, asymmetric matrices with a special structure superimposed due to the time-shift. We demonstrate that the associated eigen…

The paper proves smoothness of Brakke flows up to the end-time.

The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally oriented. There are many natural classes of examples: links of singularities; links at…

We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.

This is an expository article about groups generated by two isometries of the complex hyperbolic plane.

We construct a compact nonpositively curved squared 2-complex whose universal cover contains a flat plane that is not the limit of periodic flat planes.

Proofs Lie's classification of certain vector field subalgebras.

An isomorphism of symplectically tame smooth pseudocomplex structures on the complex projective plane which is a homeomorphism and differentiable of full rank at two points is smooth.

Every lens space has a locally flat embedding in a connected sum of 8 copies of the complex projective plane and a smooth embedding in n copies of the complex projective plane for some positive integer n. We show that there is no n such that every lens space smoothly embeds in n copies of the complex projective plane.

New integral expression quantizes Arnold strangeness.

In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…

We prove Ptolemaean Inequality and Ptolemaeus' Theorem in the closure complex hyperbolic plane endowed with the Cygan metric.

The paper classifies real hypersurfaces with a specific Jacobi operator in complex Grassmannians.

Study on rational projective planes with small index singularities.

The paper classifies discrete complex hyperbolic triangle groups.