High frequency sampling of a continuous-time ARMA process

Continuous-time autoregressive moving average (CARMA) processes have recently been used widely in the modeling of non-uniformly spaced data and as a tool for dealing with high-frequency data of the form $Y_{n\Delta}, n=0,1,2,...$, where $\Delta$ is small and positive. Such data occur in many fields of application, particularly in finance and the study of turbulence. This paper is concerned with the characteristics of the process (Y_{n\Delta})_{n\in\bbz}, when $\Delta$ is small and the underlying continuous-time process (Y_t)_{t\in\bbr} is a specified CARMA process.Comment: 13 pages, submitte

The phase transition in the configuration model

Let $G=G(d)$ be a random graph with a given degree sequence $d$, such as a random $r$-regular graph where $r\ge 3$ is fixed and $n=|G|\to\infty$. We study the percolation phase transition on such graphs $G$, i.e., the emergence as $p$ increases of a unique giant component in the random subgraph $G[p]$ obtained by keeping edges independently with probability $p$. More generally, we study the emergence of a giant component in $G(d)$ itself as $d$ varies. We show that a single method can be used to prove very precise results below, inside and above the `scaling window' of the phase transition, matching many of the known results for the much simpler model $G(n,p)$. This method is a natural extension of that used by Bollobas and the author to study $G(n,p)$, itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting.Comment: 37 page

A surprising property of the least eigenvalue of a graph

AbstractLet λ(G) be the least eigenvalue of a graph G. A real number r has the induced subgraph property provided λ(G)<r implies G has an induced subgraph H with λ(H)=r. It is shown that the only numbers with the induced subgraph property are 0, −1, −2, and −2

Small Scale Retrodigitization

summary:The digitization of papers born in the print-only era is vital for the health of the mathematical record. Many large scale retrodigitization projects are underway and, at this point, probably more that half of the mathematical history has been finished. Many smaller journals and books remain to be done. This paper gives a framework within which these may also be completed. It uses the digitization of the Canadian Journal of Mathematics (53,000 pages), completed as a one-man project over a few months, as the working example. The project described herein not only may be used as a model for similar efforts but also indicates some interesting problems yet to be solved

Social Stratification and Political Behavior: An Emphasis upon Structural Dynamics

The interrelationship of social stratification factors (Social stratification is the relative position of ranks, and their distribution found within a society.) and political institutions is a frequent problem of interest to social scientists. In studies of the relationship between political behavior and social stratification, there are numerous analyses of location or rank. in the stratification system and their effects on political behavior. There also are studies on the relationship between social mobility, status crystallization, (Social mobility for the purposes of this study concerns the comparative social rank between a father and his son.) and political behavior. The interaction between mobility, crystallization, and politics has been alluded to throughout the literature, but there appear to be few systematic propositions or theories about this area. We can examine this problem in the works of learned men of many disciplines. Status crystallization is an individual\u27s consistency in rank for several status dimensions, specifically occupation, education, ethnicity, religion, and income

Social Stratification and Political Behavior: An Emphasis upon Structural Dynamics

The interrelationship of social stratification factors (Social stratification is the relative position of ranks, and their distribution found within a society.) and political institutions is a frequent problem of interest to social scientists. In studies of the relationship between political behavior and social stratification, there are numerous analyses of location or rank. in the stratification system and their effects on political behavior. There also are studies on the relationship between social mobility, status crystallization, (Social mobility for the purposes of this study concerns the comparative social rank between a father and his son.) and political behavior. The interaction between mobility, crystallization, and politics has been alluded to throughout the literature, but there appear to be few systematic propositions or theories about this area. We can examine this problem in the works of learned men of many disciplines. Status crystallization is an individual\u27s consistency in rank for several status dimensions, specifically occupation, education, ethnicity, religion, and income

Deformed Gaussian Orthogonal Ensemble description of Small-World networks

The study of spectral behavior of networks has gained enthusiasm over the last few years. In particular, Random Matrix Theory (RMT) concepts have proven to be useful. In discussing transition from regular behavior to fully chaotic behavior it has been found that an extrapolation formula of the Brody type can be used. In the present paper we analyze the regular to chaotic behavior of Small World (SW) networks using an extension of the Gaussian Orthogonal Ensemble. This RMT ensemble, coined the Deformed Gaussian Orthogonal Ensemble (DGOE), supplies a natural foundation of the Brody formula. SW networks follow GOE statistics till certain range of eigenvalues correlations depending upon the strength of random connections. We show that for these regimes of SW networks where spectral correlations do not follow GOE beyond certain range, DGOE statistics models the correlations very well. The analysis performed in this paper proves the utility of the DGOE in network physics, as much as it has been useful in other physical systems.Comment: Replaced with the revised version, accepted for publication in Phys. Rev.

Asymptotic normality of the size of the giant component via a random walk

In this paper we give a simple new proof of a result of Pittel and Wormald concerning the asymptotic value and (suitably rescaled) limiting distribution of the number of vertices in the giant component of $G(n,p)$ above the scaling window of the phase transition. Nachmias and Peres used martingale arguments to study Karp's exploration process, obtaining a simple proof of a weak form of this result. We use slightly different martingale arguments to obtain a much sharper result with little extra work.Comment: 11 pages; slightly expanded, reference adde