Moments in graphs

Let $G$ be a connected graph with vertex set $V$ and a {\em weight function} $\rho$ that assigns a nonnegative number to each of its vertices. Then, the {\em $\rho$-moment} of $G$ at vertex $u$ is defined to be M_G^{\rho}(u)=\sum_{v\in V} \rho(v)\dist (u,v) , where \dist(\cdot,\cdot) stands for the distance function. Adding up all these numbers, we obtain the {\em $\rho$-moment of $G$}: M_G^{\rho}=\sum_{u\in V}M_G^{\rho}(u)=1/2\sum_{u,v\in V}\dist(u,v)[\rho(u)+\rho(v)]. This parameter generalizes, or it is closely related to, some well-known graph invariants, such as the {\em Wiener index} $W(G)$, when $\rho(u)=1/2$ for every $u\in V$, and the {\em degree distance} $D'(G)$, obtained when $\rho(u)=\delta(u)$, the degree of vertex $u$. In this paper we derive some exact formulas for computing the $\rho$-moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding $\rho$-moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same $\rho$-moment for every $\rho$ (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product

Accounting for changes in the homeownership rate

After three decades of being relatively constant, the homeownership rate increased over the 1994–2005 period to attain record highs. The objective of this paper is to account for the observed boom in ownership by examining the role played by changes in demographic factors and innovations in the mortgage market that lessened down payment requirements. To measure the aggregate and distributional impact of these factors, we construct a quantitative general equilibrium overlapping-generation model with housing. We find that the long-run importance of the introduction of new mortgage products for the aggregate homeownership rate ranges from 56 percent to 70 percent. Demographic factors account for between 16 percent and 31 percent of the change. Transitional analysis suggests that demographic factors play a more important but not dominant role farther from the long-run equilibrium. From a distributional perspective, mortgage market innovations have a larger impact on participation rate changes of younger households, and demographic factors seem to be the key to understanding the participation rate changes of older households. Our analysis suggests that the key to understanding the increase in the homeownership rate is the expansion of the set of mortgage contracts. We test the robustness of this result by considering changes in mortgage financing after World War II. We find that the introduction of the conventional fixed-rate mortgage, which replaced balloon contracts, accounts for at least 50 percent of the observed increase in homeownership during that period.

"Accounting for Changes in the Homeownership Rate"

After years of being relatively constant, the homeownership rate -a target for housing policy- has increased since 1995. This paper attempts to understand why the homeownership rate has been increasing by constructing a quantitative model and then using this model to evaluate explanations that have been offered to account for this increase. We find that the increase in the homeownership can be explained by innovations in the mortgage market that allows households to take a positive housing investment position with a much smaller downpayment.

Holographic multiverse and the measure problem

We discuss the duality, conjectured in earlier work, between the wave function of the multiverse and a 3D Euclidean theory on the future boundary of spacetime. In particular, we discuss the choice of the boundary metric and the relation between the UV cutoff scale xi on the boundary and the hypersurfaces Sigma on which the wave function is defined in the bulk. We propose that in the limit of xi going to 0 these hypersurfaces should be used as cutoff surfaces in the multiverse measure. Furthermore, we argue that in the inflating regions of spacetime with a slowly varying Hubble rate H the hypersurfaces Sigma are surfaces of constant comoving apparent horizon (CAH). Finally, we introduce a measure prescription (called CAH+) which appears to have no pathological features and coincides with the constant CAH cutoff in regions of slowly varying H.Comment: A minor change: the discussion of unitarity on p.9 is clarifie

Energy shaping control of soft continuum manipulators with in-plane disturbances

Soft continuum manipulators offer levels of compliance and inherent safety that can render thema superior alternative to conventional rigid robotsfor a variety of tasks, such as medical interventions or human-robot interaction. However, the ability of soft continuum manipulators to compensate external disturbances need to be further enhanced to meet the stringent requirements of many practical applications.In this paper, we investigate the control problem forsoft continuum manipulators that consist of one inextensible segmentof constant section, which bends under the effect of the internal pressure and is subject to unknown disturbances acting in the plane of bending. A rigid-link model of the manipulatorwith a single input pressureis employed for control purposes and an energy-shaping approach isproposedto derive thecontrol law. A method for the adaptive estimation of disturbances is detailed and a disturbance compensation strategy is proposed.Finally, the effectiveness of the controlleris demonstrated with simulations and with experiments on an inextensible soft continuum manipulator that employs pneumatic actuation

On Almost Distance-Regular Graphs

2010 Mathematics Subject Classification: 05E30, 05C50;distance-regular graph;walk-regular graph;eigenvalues;predistance polynomial

Number of walks and degree powers in a graph

This note deals with the relationship between the total number of $k$-walks in a graph, and the sum of the $k$-th powers of its vertex degrees. In particular, it is shown that the the number of all $k$-walks is upper bounded by the sum of the $k$-th powers of the degrees

Action of Singular Instantons of Hawking-Turok Type

Using Kaluza-Klein technique we show that the singularity of Hawking-Turok type has a fixed point (bolt) contribution to the action in addition to the usual boundary contribution. Interestingly by adding this contribution we can obtain a simple expression for the total action which is feasible for both regular and singular instantons. Our result casts doubt on the constraint proposed by Turok in the recent calculation in which Vilenkin's instantons are regarded as a limit of certain constrained instantons.Comment: 14 pages, LaTe