arXiv research

Decomposes axis bundles into cubist structures for fully irreducible outer automorphisms.

Isothermic nets created from special maps for smooth surfaces.

This paper explores non-periodic folding of Spidron units, revealing nonlinear dynamics.

The paper describes the geometric properties of line congruences' singularities.

Let K be a connected finite complex. This paper studies the problem of whether one can attach a cell to some iterated suspension S^j K so that the resulting space satisfies Poincare duality. When this is possible, we say that S^j K is a spine. We introduce the notion of quadratic self duality and show that if K is quad…

New multi-spiral approach improves packaging of thick membranes.

H-holomorphic maps are a parameter version of J-holomorphic maps into contact manifolds. They have arisen in efforts to prove the existence of higher--genus holomorphic open book decompositions and efforts to prove the existence of finite energy foliations and the Weinstein conjecture, as well as in folded holomorphic …

We are introducing a general framework for the construction of polyhedra and simplicial comlexes that are {\em bifoldable}, i.e. foldable into two two different planes. This vastly generalizes Origami folds known as the Miura pattern, the Eggbox pattern. After describing the framework and its basic features, we give se…

We construct periodic families of Poincare complexes, partially solving a question of Hodgson that was posed in the proceedings of the 1982 Northwestern homotopy theory conference. We also construct infinite families of Poincare complexes whose top cell falls off after one suspension but which fail to embed in a sphere…

In this work we generalize the classical notion of a (compact) twistor line in the period domain of compact complex tori. We introduce two new types of lines, which are non-compact analytic curves in the period domain of complex tori. We study the analytic properties of the compactifications of the curves, the preserva…

Calculates Laplacian spectra on Calabi-Yau hypersurfaces.

We establish the general formalism for constructing metrics of Calabi-Yau (p+1)-folds in terms of that of a p-fold by adding a complex-line bundle. We present a few explicit low-lying examples. We further consider holomorphic linearization and obtain the six-dimensional analogue of the Gibbons-Hawking instanton. Whilst…

In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R^3. By means of a connection of the differential equations for these curvature lines and real Ricc…

New minimal surfaces found with spherical curvature lines.

We derive new obstructions to periodicity of classical knots by employing the Heegaard Floer correction terms of the finite cyclic branched covers of the knots. Applying our results to two fold covers, we demonstrate through numerous examples that our obstructions are successful where many existing periodicity obstruct…

As in the case of irreducible holomorphic symplectic manifolds, the period domain $Compl$ of compact complex tori of even dimension $2n$ contains twistor lines. These are special $2$-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irredu…

Paper proves structure of compact Kähler 3-folds with specific bundles.

Study shows invariant curves in tubular origami dynamics, revealing geometric barriers to folding transitions.

Study of straight-line flows on a unique infinite surface.

The paper explores reflection principles for lightlike line segments on maximal surfaces.

Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of …

For any integer $k$ we construct an explicit example of a twistor space which contains a one--parameter family of jumping rational curves, where the normal bundle changes from $O(1)+O(1)$ to $O(k)+O(2-k)$. For $k>3$ the resulting anti--self--dual Ricci-flat manifold is a Zariski cone in the space of holomorphic section…

The paper explores the geometry of level lines of quasiperiodic functions with many periods.

We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in {\it Bulletin of the AMS}, 29(1):77--84, 1993. Its ends in the quotient are asymptotic to one full turn of the …

Study k-folding map-germs to understand surface geometry.

The study examines the stretch factors of outer automorphisms and their latent symmetry.

This is the fourth in a series of papers math.DG/0008021, math.DG/0008155, math.DG/0010036 constructing explicit examples of special Lagrangian submanifolds (SL m-folds) in C^m. A submanifold of C^m is ruled if it is fibred by a family of real straight lines in C^m. This paper studies ruled special Lagrangian 3-folds i…

Computes canonical heights for arithmetic log surfaces using Hurwitz zeta function.

One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental inte…

Study finds periodic orbits in a complex gravitational system.

We study quotients $Γ\backslash \mathbb H^n$ of the $n$-fold product of the upper half plane $\mathbb H$ by irreducible and torsion-free lattices $Γ< PSL_2(\mathbb R)^n$ with the same Betti numbers as the $n$-fold product $(\mathbb P^1)^n$ of projective lines. Such varieties are called fake products of projective lines…

Python tool detects economic crises from S&P500 correlation data.

We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of correspondin…

We propose a reduced form set of two coupled continuous time equations linking the price of a representative asset and the price of a bond, the later quantifying the cost of borrowing. The feedbacks between asset prices and bonds are mediated by the dependence of their "fundamental values" on past asset prices and bond…

Study classifies stock price data into stationary and non-stationary periods for mechanical trading.

In this work a spectral theory for 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation is developed. Spectral data for such solutions are defined (following Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic …

Researchers find metric lines in SE(2) using Hamilton-Jacobi theory.

Eisenbud Popescu and Walter have constructed certain special 4-dimensional sextic hypersurfaces as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3-surface and that the family of such varieties is locally complete for deformations …

We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an ana…

On-line portfolio selection has attracted increasing interests in machine learning and AI communities recently. Empirical evidences show that stock's high and low prices are temporary and stock price relatives are likely to follow the mean reversion phenomenon. While the existing mean reversion strategies are shown to …

New methods create full discretized isothermic tori in Euclidean spaces.

Study analyzes stock performance before, during, and after the pandemic.

The straight-line flow on almost every staircase and on almost every square tiled staircase is recurrent. For almost every square tiled staircase the set of periodic orbits is dense in the phase space.

Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.

This paper establishes that the Nahm transform sending spatially periodic instantons (instantons on the product of the real line and a three-torus) to singular monopoles on the dual three-torus is indeed a bijection as suggested by the heuristic. In the process, we show how the Nahm transform intertwines to a Fourier-M…

The simplest patterns of qualitative changes on the configurations of lines of principal curvature} around umbilic points on surfaces whose immersions into $\mathbb R^3$ depend smoothly on a real parameter (codimension one umbilic bifurcations) are described in this paper. Global effects, due to umbilic bifurcations, o…

We construct an obstruction for the existence of embeddings of homology $3$-sphere into homology $S^3\times S^1$ under some cohomological condition. The obstruction is defined as an element in the filtered version of the instanton Floer cohomology due to R.Fintushel-R.Stern. We make use of the $\mathbb{Z}$-fold coverin…

The current article studies certain problems related to complex cycles of holomorphic foliations with singularities in the complex plane. We focus on the case when polynomial differential one-form gives rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the f…