arXiv research

Improved theorem on curvature and manifold symmetry.

Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.

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…

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…

This paper investigates arbitrage chains involving four currencies and four foreign exchange trader-arbitrageurs. In contrast with the three-currency case, we find that arbitrage operations when four currencies are present may appear periodic in nature, and not involve smooth convergence to a "balanced" ensemble of exc…

We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S^1 spaces. Additionally, we show that all these spaces are Kaehler, that every such space is obtained from a simple model by a sequence of symplectic blowups, and that if the fixed points are is…

For a real valued periodic smooth function u on R, $n\ge 0$, one defines the osculating polynomial $φ_s$ (of order 2n+1) at a point $s\in R$ to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex …

No time-periodic Majorana fermions found in Kerr-Newman spacetimes with nontrivial charge.

The paper proves the existence of maxfaces with multiple swallowtails and planar ends.

We extend the Atiyah, Patodi, and Singer index theorem for first order differential operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. This theorem provides a natural complement to Taubes' Fredholm theory for general end-periodic operators. Our index theorem is expressed in t…

Constructs a family of genus three minimal surfaces with parallel ends.

Proofs for Moon's theorem and its generalization.

We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah-Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our theorem with a classical result of Salamon and Zehnder. Finally, we use the index theo…

New proof of four squares theorem using projective geometry.

Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold is determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calcula…

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 3-dimensional Euclidean space, Scherk second surfaces are singly periodic embedded minimal surfaces with four planar ends. In this paper, we obtain a natural generalization of these minimal surfaces in any higher dimensional Euclidean space ${\R}^{n+1}$, for $n \geq 3$. More precisely, we show that there exist $(n-1…

We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain of 3-space with the same volume. The minimizer is the so-called srs network with quotient the complete graph on four vertices $K_4$. The network…

We construct Weierstrass data for higher genus embedded doubly periodic minimal surfaces and present numerical evidence that the associated period problem can be solved. In the orthogonal ends case, there previously was only one known surface for each genus. We illustrate multiple new examples for each genus g>2. In th…

Extends Adams' theorem to periodic cohomology.

Study finds how periodic surfaces can bend without stretching.

The study of rigidity theorems on 4-manifolds with boundary.

We refine Theorem A due to Gursky \cite{G3}. As applications, we give some rigidity theorems on four-manifolds with postive Yamabe constant. In particular, these rigidity theorems are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of e…

Relative moduli spaces of periodic monopoles provide novel examples of Asymptotically Locally Flat hyperkahler manifolds. By considering the interactions between well-separated periodic monopoles, we infer the asymptotic behavior of their metrics. When the monopole moduli space is four-dimensional, this construction yi…

Simple proof shows forecasts can be calibrated in a few periods.

Since their introduction by Thurston, geodesic laminations on hyperbolic surfaces occur in many contexts. In this paper, we propose a generalization of geodesic laminations on locally CAT(0), complete, geodesic metric spaces, whose boundary at infinity of the universal cover is endowed with a invariant total cyclic ord…

The study proves a discrete version of Segre's theorem for polygonal curves.

Minimal surfaces span periodic curves in 3D space.

Classifies minimal surfaces of finite genus in 3D space.

Hypothesis of Market Efficiency is an important concept for the investors across the globe holding diversified portfolios. With the world economy getting more integrated day by day, more people are investing in global emerging markets. This means that it is pertinent to understand the efficiency of these markets. This …

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.

Proves a theorem for mechanical systems with reflections.

We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell-Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorit…

The minute-by-minute move of the Hang Seng Index (HSI) data over a four-year period is analysed and shown to possess similar statistical features as those of other markets. Based on a mathematical theorem [S. B. Pope and E. S. C. Ching, Phys. Fluids A {\bf 5}, 1529 (1993)], we derive an analytic form for the probabilit…

The singly periodic genus-one helicoid was in the origin of the discovery of the first example of a complete minimal surface with finite topology but infinite total curvature, the celebrated Hoffman-Karcher-Wei's genus one helicoid. The objective of this paper is to give a uniqueness theorem for the singly periodic gen…

The period map for 4-manifolds is dense and surjective under certain conditions.

Maps minimize periodic points for high periods, but not for low periods.

In this paper, we prove a classification theorem of 4-manifolds according to some conformal invariants, which generalizes the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}. Moreover, it provides a four-dimensional analogue of the well-known classification theorem of Schoen-Yau \cite{SY2} on 3-man…

New findings on knot genera using advanced techniques.

This article has two purposes. The first is to give an expository account of the integrable systems approach to harmonic maps from surfaces to Lie groups and symmetric spaces, focusing on spectral curves for harmonic 2-tori. The most unwieldy aspect of the spectral curve description is the periodicity conditions and th…

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proo…

In this paper, we address the issue of quaternionic Toledo invariant to study the character variety of two dimensional complex hyperbolic uniform lattices into $SU(n,2)$. We construct four distinct representations to prove that the character variety contains at least four distinct components. We also address the existe…

In this article we present a method by which we can reduce a time series into a single point in $\mathbb{R}^{13}$. We have chosen 13 dimensions so as to prevent too many points from being labeled as "noise." When using a Euclidean (or Mahalanobis) metric, a simple clustering algorithm will with near certainty label the…

Proves K-K-W type theorems for specific types of operators.

Li-York theorem tells us that a period 3 orbit for a continuous map of the interval into itself implies the existence of a periodic orbit of every period. This paper concerns an analogue of the theorem for homeomorphisms of the 2-dimensional disk. In this case a periodic orbit is specified by a braid type and on the se…

The study examines vertices in curves with singular points in the Euclidean plane.

The paper proves unique characterization of gravitational instantons with specific volume growth.

Given a three-manifold with b_1=1 and a nontorsion spin^c structure, we use finite dimensional approximation to construct from the Seiberg-Witten equations two invariants in the form of a periodic pro-spectra. Various functors applied to these invariants give different flavors of Seiberg-Witten Floer homology. We also …