Physical Sciences and Mathematics Commons^™

An Indefinite Kähler Metric On The Space Of Oriented Lines, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space T of oriented affine lines in R3 with the tangent bundle of S2. Thus the round metric on S2 induces a Kähler structure on T which turns out to have a metric of neutral signature. It is shown that the identity component of the isometry group of this metric is isomorphic to the identity component of the isometry group of the Euclidean metric on R3.

The geodesics of this metric are either planes or helicoids in R3. The signature …

Synchrony In A Boolean Network Of The L-Arabinose Operon In Escherichia Coli, Andy Jenkins, Matthew Macauley

Publications

The lactose operon in Escherichia coli was the first known gene regulatory network, and it is frequently used as a prototype for new modeling paradigms. Historically, many of these modeling frameworks use differential equations. More recently, Stigler and Veliz-Cuba proposed a Boolean network model that captures the bistability of the system and all of the biological steady states. In this paper, we model the well-known arabinose operon in E. coli with a Boolean network. This has several complex features not found in the lac operon, such as a protein that is both an activator and repressor, a DNA looping mechanism …

A Converging Lagrangian Flow In The Space Of Oriented Lines, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the three-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is the Steiner point of the initial surface, which remains constant under the flow.
To prove this we show that the oriented normal lines, considered as a surface in the space of all oriented lines, evolve by a parabolic flow which preserves the Lagrangian condition.Moreover, this flow converges to a holomorphic Lagrangian section, which forms …

Noncrossing Partitions, Toggles, And Homomesies, David Einstein, Miriam Farber, Emily Gunawan, Michael Joseph, Matthew Macauley, James Propp, Simon Rubinstein-Salzedo

Publications

We introduce n(n−1)/2 natural involutions (“toggles”) on the set S of noncrossing partitions π of size n, along with certain composite operations obtained by composing these involutions. We show that for many operations T of this kind, a surprisingly large family of functions f on S (including the function that sends π to the number of blocks of π) exhibits the homomesy phenomenon: the average of f over the elements of a T -orbit is the same for all T -orbits. We can apply our method of proof more broadly to toggle operations back on the collection of independent sets …

Ermakov Equation And Camassa-Holm Waves, Haret C. Rosu, S.C. Mancas

Publications

From the works of authors of this article, it is known that the solution of the Ermakov equation is an important ingredient in the spectral problem of the Camassa-Holm equation. Here, we review this interesting issue and consider in addition more features of the Ermakov equation which have an impact on the behavior of the shallow water waves as described by the Camassa-Holm equation.

Totally Null Surfaces In Neutral K¨Ahler 4-Manifolds, Nikos Georgiou, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

We study the totally null surfaces of the neutral K¨ahler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual (α-planes) or anti-self-dual (β-planes) and so we consider α-surfaces and β-surfaces. The metric of the examples we study, which include the spaces of oriented geodesics of 3-manifolds of constant curvature, are anti-self-dual, and so it is well-known that the α-planes are integrable and α-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The β-surfaces are less known and our interest is mainly in their …

Morphisms And Order Ideals Of Toric Posets, Matthew Macauley

Publications

Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper, we define toric intervals and toric order-preserving maps, which lead to toric analogues of poset morphisms and order ideals. We develop this theory, discuss some fundamental differences between the toric and ordinary cases, …

Integrable Abel Equations And Vein's Abel Equation, S.C. Mancas, Haret C. Rosu

Publications

We first reformulate and expand with several novel findings some of the basic results in the integrability of Abel equations. Next, these results are applied to Vein’s Abel equation whose solutions are expressed in terms of the third order hyperbolic functions and a phase space analysis of the corresponding nonlinear oscillator is also provided.

What Moser Could Have Asked: Counting Hamilton Cycles In Tournaments, Neil J. Calkin, Beth Novick, Hayato Ushijima-Mwesigwa

Publications

Moser asked for a construction of explicit tournaments on n vertices having at least Hamilton cycles. We show that he could have asked rather more.

Relaxations And Discretizations For The Pooling Problem, Akshay Gupte, Shabbir Ahmed, Santanu S. Dey, Myun Seok Cheon

Publications

The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives new insights on prevalent techniques. We also present new ideas for computing dual bounds on the global optimum by solving high-dimensional linear programs. Finally, we propose discretization methods for inner approximating the feasible region and obtaining good primal bounds. Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs. The strength of our relaxations …

Nongauge Bright Soliton Of The Nonlinear Schrodinger (Nls) Equation And A Family Of Generalized Nls Equations, M. A. Reyes, D. Gutierrez-Ruiz, S. C. Mancas, H. C. Rosu

Publications

We present an approach to the bright soliton solution of the nonlinear Schrödinger (NLS) equation from the standpoint of introducing a constant potential term in the equation. We discuss a “nongauge” bright soliton for which both the envelope and the phase depend only on the traveling variable. We also construct a family of generalized NLS equations with solitonic sechpsechp solutions in the traveling variable and find an exact equivalence with other nonlinear equations, such as the Korteveg–de Vries (KdV) and Benjamin–Bona–Mahony (BBM) equations when p=2.

Existence Of Periodic Orbits In Nonlinear Oscillators Of Emden-Fowler Form, S.C. Mancas, Haret C. Rosu

Publications

The nonlinear pseudo-oscillator recently tackled by Gadella and Lara is mapped to an Emden–Fowler (EF) equation that is written as an autonomous two-dimensional ODE system for which we provide the phase-space analysis and the parametric solution. Through an invariant transformation we find periodic solutions to a certain class of EF equations that pass an integrability condition. We show that this condition is necessary to have periodic solutions and via the ODE analysis we also find the sufficient condition for periodic orbits. EF equations that do not pass integrability conditions can be made integrable via an invariant transformation which also allows …

A Note On Vector Valued Discrete Schrödinger Operators, Keshav R. Acharya

Publications

The main purpose of this paper is to extend some theory of Schrödinger operators from one dimension to higher dimension. In particular, we will give systematic operator theoretic analysis for the Schrödinger equations in multidimensional space. To this end, we will provide the detail proves of some basic results that are necessary for further studies in these areas. In addition, we will introduce Titchmarsh- Weyl m− function of these equations and express m− function in term of the resolvent operators.

Remling's Theorem On Canonical Systems, Keshav R. Acharya

Publications

In this paper, we extend the Remling’s Theorem on canonical systems that the ω limit points of the Hamiltonian under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure of a canonical system.