Path count asymptotics and Stirling numbers

We obtain formulas for the growth rate of the numbers of certain paths in infinite graphs built on the two-dimensional Eulerian graph. Corollaries are identities relating Stirling numbers of the first and second kinds.Comment: Misprint corrected. To appear in Proc. Amer. Math. So

A Hedonic Model for Housing Prices in Wilsonville, Oregon

We estimate a hedonic model for housing prices in Wilsonville, Oregon. Our data for 197 houses is drawn from Zillow for 2014 to 2017. We find that the number of bedrooms, the square footage of the house, and whether the house is single level are statistically significant factors affecting housing prices. Location variables are also found to be statistically significant. As an example, the price of a house on the Charbonneau Golf Course is estimated to be 15% higher than the price of a house located elsewhere in Wilsonville

Counting Dyck paths by area and rank

The set of Dyck paths of length $2n$ inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths: \emph{area} (the area under the path) and \emph{rank} (the rank in the lattice). While area for Dyck paths has been studied, pairing it with this rank function seems new, and we get an interesting $(q,t)$-refinement of the Catalan numbers. We present two decompositions of the corresponding generating function: one refines an identity of Carlitz and Riordan; the other refines the notion of $\gamma$-nonnegativity, and is based on a decomposition of the lattice of noncrossing partitions due to Simion and Ullman. Further, Biane's correspondence and a result of Stump allow us to conclude that the joint distribution of area and rank for Dyck paths equals the joint distribution of length and reflection length for the permutations lying below the $n$-cycle $(12...n)$ in the absolute order on the symmetric group.Comment: 24 pages, 7 figures. Connections with work of C. Stump (arXiv:0808.2822v2) eliminated the need for 5 pages of proof in the first draf

Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures with Free Body Motion

Negative imaginary (NI) systems play an important role in the robust control of highly resonant flexible structures. In this paper, a generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors, and with free body motion. This definition extends the existing definitions of NI systems. Also, necessary and sufficient conditions are provided for the stability of positive feedback control systems where the plant is NI according to the new definition and the controller is strictly negative imaginary. The stability conditions in this paper are given purely in terms of properties of the plant and controller transfer function matrices, although the proofs rely on state space techniques. Furthermore, the stability conditions given are independent of the plant and controller system order. As an application of these results, a case study involving the control of a flexible robotic arm with a piezo-electric actuator and sensor is presented

Nonexistence results for the Korteweg-deVries and Kadomtsev-Petviashvili equations

We study characteristic Cauchy problems for the Korteweg-deVries (KdV) equation $u_t=uu_x+u_{xxx}$, and the Kadomtsev-Petviashvili (KP) equation $u_{yy}=\bigl(u_{xxx}+uu_x+u_t\bigr)_x$ with holomorphic initial data possessing nonnegative Taylor coefficients around the origin. For the KdV equation with initial value $u(0,x)=u_0(x)$, we show that there is no solution holomorphic in any neighbourhood of $(t,x)=(0,0)$ in ${\mathbb C}^2$ unless $u_0(x)=a_0+a_1x$. This also furnishes a nonexistence result for a class of $y$-independent solutions of the KP equation. We extend this to $y$-dependent cases by considering initial values given at $y=0$, $u(t,x,0)=u_0(x,t)$, $u_y(t,x,0)=u_1(x,t)$, where the Taylor coefficients of $u_0$ and $u_1$ around $t=0$, $x=0$ are assumed nonnegative. We prove that there is no holomorphic solution around the origin in ${\mathbb C}^3$ unless $u_0$ and $u_1$ are polynomials of degree 2 or lower.Comment: 17 pages in LaTeX2e, to appear in Stud. Appl. Mat

Nonsquare Spectral Factorization for Nonlinear Control Systems

This paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time. More specifically, we obtain a parametrization of nonsquare spectral factors in terms of invariant Lagrangian submanifolds and associated solutions of Hamilton–Jacobi inequalities. This inequality is a nonlinear analogue of the bounded real lemma and the control algebraic Riccati inequality. By way of an application, we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system.