On the Smallest Eigenvalue of Grounded Laplacian Matrices

We provide upper and lower bounds on the smallest eigenvalue of grounded Laplacian matrices (which are matrices obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between the upper and lower bounds depends on the ratio of the smallest and largest components of the eigenvector corresponding to the smallest eigenvalue of the grounded Laplacian. We provide a graph-theoretic bound on this ratio, and subsequently obtain a tight characterization of the smallest eigenvalue for certain classes of graphs. Specifically, for Erdos-Renyi random graphs, we show that when a (sufficiently small) set $S$ of rows and columns is removed from the Laplacian, and the probability $p$ of adding an edge is sufficiently large, the smallest eigenvalue of the grounded Laplacian asymptotically almost surely approaches $|S|p$. We also show that for random $d$-regular graphs with a single row and column removed, the smallest eigenvalue is $\Theta(\frac{d}{n})$. Our bounds have applications to the study of the convergence rate in continuous-time and discrete-time consensus dynamics with stubborn or leader nodes

Microeconomic Inventory Behavior and Aggregate Inventory Dynamics

The slow adjustment of inventory stocks to changes in sales has been a puzzle for the inventory literature since at least Auerbach and Feldstein (1976). Recent evidence suggests that estimated firm-level adjustment speeds of inventory stocks are significantly higher than estimates based on aggregate data. This paper investigates the circumstances under which such bias occurs using an industry equilibrium model where, consistently with empirical evidence, some firms smooth production while others bunch it. The model can account for the significant downward bias documented empirically when a subset of firms displays countercyclical mark-up movements.

Competition and cooperation in a metal engineering production system

In the discussion on the prospects for growth of the manufacturing system in Italy one still unsolved problem stands out: the small size of the firms. There is a great concern regarding not only sectors facing strong competition from countries with a low labour cost, but even sectors with a good position in the world market, as the engineering firms in the province of Modena. As a matter of fact, in the mechanical-engineering sector there is a large number of small firms and only very few firms belong to “groups” (and instances of foreign groups are rare): small size of independent companies is considered a sign of weakness that could be a mark of their inability to operate on international markets and thus to face the challenges of globalization. The paper investigates the systemic characteristic of the mechanical-engineering production system in Modena and the strength of many short chains of linkages within the network of companies operating at local level for the global markets. Our focus is the dynamics of change of the system. The literature on industrial districts has frequently emphasized how the firms that operate in the district are in competition with one another, when it is a question of firms specializing in the same stage of the production process; whereas they cooperate in the case of firms operating in different stages in the same production filière. This particular pattern of competition and cooperation among firms specializing in a stage could be one of the distinguishing marks of the system (“equilibrium” factors, as Brusco, 1989 and 1999, calls them). This explanation supposes that the firms can be either in competition or cooperating, we find forms of competition, for certain activities, among firms that cooperate for other activities. The data on the presence of competitors among the suppliers or the clients give an idea of how extensive this phenomenon is in the Modena engineering system. In this paper we show that the weak points of Modena’s mechanical-engineering industry lie not so much in the size of the firm as in the mechanisms that fuel and regenerate the competences needed to sustain the development of the network of firms. This line of research opens new question in the analysis of market systems and network of competences that are addressed in the last part of the paper.local production system, mechanical-engineering firms,cooperation, competition, market system

Therapeutic encounters at a Muslim shrine in Pakistan: an ethnographic study of understandings and explanations of ill health and help-seeking among attenders.

In Muslim countries, shrines of Sufi saints serve as sources of healing. Why people decide to seek healing at shrines and their experience whilst there remains largely un-researched. The aims of this study were to: investigate the explanatory models of sickness among attendees at a Muslim shrine particularly in relation to this choice of help-seeking; explore individuals’ perceptions and experiences regarding the role of the shrine; and propose a theory explaining the meaning of attendees’ problems, their choice of healing resource and the role it played. An ethnographic approach was used to allow exploration of the topic from the perspectives of those seeking help at the Shrine. Semi-structured interview, incorporating the Explanatory Model (EM) of sickness (Kleinman, 1980), and participant observation were used to collect data over a period of three months. This study was conducted at a Muslim shrine in Pakistan. Twenty six attendees participated, including those seeking healing, carers, and a Shrine caretaker. The results highlighted magic and possession as the main explanations of the problem that brought them to the Shrine. Participants’ experiences of everyday oppression, and adverse social factors, such as poverty, poor quality of medical care, and domestic violence seemed to play a significant role in the development of their problem. The Shrine served as a therapeutic landscape, the prevailing social conditions, built environment and perceptions of attendees combined to produce a place ‘conducive to healing’ (Gesler, 1992), that allowed healing to take place. Possession acted as a vehicle for a subtle change in the family dynamics in which family members appeared complicit. The movement and changes in power, the positive reframing of symptoms/problems and the renegotiation of identity essentially transformed the individual and made the experience therapeutic. The results generate a unique set of knowledge in regard to the role of shrines in Pakistan as culturally sanctioned places allowing therapeutic change and healing

Gravitational waves in vacuum spacetimes with cosmological constant. II. Deviation of geodesics and interpretation of non-twisting type N solutions

In a suitably chosen essentially unique frame tied to a given observer in a general spacetime, the equation of geodesic deviation can be decomposed into a sum of terms describing specific effects: isotropic (background) motions associated with the cosmological constant, transverse motions corresponding to the effects of gravitational waves, longitudinal motions, and Coulomb-type effects. Conditions under which the frame is parallelly transported along a geodesic are discussed. Suitable coordinates are introduced and an explicit coordinate form of the frame is determined for spacetimes admitting a non-twisting null congruence. Specific properties of all non-twisting type N vacuum solutions with cosmological constant Lambda (non-expanding Kundt class and expanding Robinson-Trautman class) are then analyzed. It is demonstrated that these spacetimes can be understood as exact transverse gravitational waves of two polarization modes "+" and "x", shifted by pi/4, which propagate "on" Minkowski, de Sitter, or anti-de Sitter backgrounds. It is also shown that the solutions with Lambda>0 may serve as exact demonstrations of the cosmic "no-hair" conjecture in radiative spacetimes with no symmetry.Comment: 16 pages, no figures, LaTeX, To appear in J. Math. Phy

Capital Ownership Under Incomplete Markets: Does it Matter?

In the macroeconomic literature, the implications of a context with household heterogeneity and incomplete financial markets have been mostly studied under the assumption that households own the physical capital and undertake the intertemporal investment decision. Further, firms rent capital and labor from the households to maximize period profits. The present paper provides the conditions under which this assumption is still irrelevant when markets are incomplete. It shows that, if firms own the physical capital and undertake the investment decision to maximize their asset value, in the sense that they discount future cash flows with positive state price processes that are consistent with security prices, the equilibrium allocations are the same as in the standard setting with static firms. On the other hand, the firm valuation of future cash flows only coincides with the valuation of the unconstrained shareholders. Given this, value maximization may still lead to shareholder disagreement in the presence of effectively binding portfolio restrictions.

A Canonical Decomposition in Collective and Relative Variables of a Klein-Gordon Field in the Rest-Frame Wigner-Covariant Instant Form

The canonical decomposition of a real Klein-Gordon field in collective and relative variables proposed by Longhi and Materassi is reformulated on spacelike hypersurfaces. This allows to obtain the complete canonical reduction of the system on Wigner hyperplanes, namely in the rest-frame Wigner-covariant instant form of dynamics. From the study of Dixon's multipoles for the energy-momentum tensor on the Wigner hyperplanes we derive the definition of the canonical center-of-mass variable for a Klein-Gordon field configuration: it turns out that the Longhi-Materassi global variable should be interpreted as a center of phase of the field configuration. A detailed study of the kinematical "external" and "internal" properties of the field configuration on the Wigner hyperplanes is done. The construction is then extended to charged Klein-Gordon fields: the centers of phase of the two real components can be combined to define a global center of phase and a collective relative variable describing the action-reaction between the two Feshbach-Villars components of the field with definite sign of energy and charge. The Dixon multipoles for both the energy-momentum and the electromagnetic current are given. Also the coupling of the Klein-Gordon field to scalar relativistic particles is studied and it is shown that in the reduced phase space, besides the particle and field relative variables, there is also a collective relative variable describing the relative motion of the particle subsytem with respect to the field one.Comment: 86 pages, no figure