Mean-Field Spin Glass models from the Cavity--ROSt Perspective

The Sherrington-Kirkpatrick spin glass model has been studied as a source of insight into the statistical mechanics of systems with highly diversified collections of competing low energy states. The goal of this summary is to present some of the ideas which have emerged in the mathematical study of its free energy. In particular, we highlight the perspective of the cavity dynamics, and the related variational principle. These are expressed in terms of Random Overlap Structures (ROSt), which are used to describe the possible states of the reservoir in the cavity step. The Parisi solution is presented as reflecting the ansatz that it suffices to restrict the variation to hierarchal structures which are discussed here in some detail. While the Parisi solution was proven to be correct, through recent works of F. Guerra and M. Talagrand, the reasons for the effectiveness of the Parisi ansatz still remain to be elucidated. We question whether this could be related to the quasi-stationarity of the special subclass of ROSts given by Ruelle's hierarchal `random probability cascades' (also known as GREM).Comment: Based on talks given at `Young Res. Symp.', Lisbon 2003, and `Math. Phys. of Spin Glasses', Cortona 200

Twisted C*-algebras associated to finitely aligned higher-rank graphs

We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish versions of the usual uniqueness theorems and the classification of gauge-invariant ideals. We show that all twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs are nuclear and satisfy the UCT, and that for twists that lift to real-valued cocycles, the K-theory of a twisted relative Cuntz-Krieger algebra is independent of the twist. In the final section, we identify a sufficient condition for simplicity of twisted Cuntz-Krieger algebras associated to higher-rank graphs which are not aperiodic. Our results indicate that this question is significantly more complicated than in the untwisted setting.Comment: Version 2: This paper has now appeared in Documenta Mathematica. This version on arXiv exactly matches the pagination and format of the published version. Original published version available from http://www.math.uni-bielefeld.de/documenta/vol-19/28.htm

Estimating the Firm's Labor Supply Curve in a "New Monopsony" Framework: School Teachers in Missouri

In the context of certain dynamic models, it is possible to infer the elasticity of labor supply to the firm from the elasticity of the quit rate with respect to the wage. Using this property, we estimate the average labor supply elasticity to public school districts in Missouri. We take advantage of the plausibly exogenous variation in pre-negotiated district salary schedules to instrument for actual salary. Instrumental variables estimates lead to a labor supply elasticity estimate of about 3.7, suggesting the presence of significant market power for school districts, especially over more experienced teachers. The presence of monopsony power in this labor market may be partially explained by institutional features of the teacher labor market.labor monopsony, teachers

Purposive discovery of operations

The Generate, Prune & Prove (GPP) methodology for discovering definitions of mathematical operators is introduced. GPP is a task within the IL exploration discovery system. We developed GPP for use in the discovery of mathematical operators with a wider class of representations than was possible with the previous methods by Lenat and by Shen. GPP utilizes the purpose for which an operator is created to prune the possible definitions. The relevant search spaces are immense and there exists insufficient information for a complete evaluation of the purpose constraint, so it is necessary to perform a partial evaluation of the purpose (i.e., pruning) constraint. The constraint is first transformed so that it is operational with respect to the partial information, and then it is applied to examples in order to test the generated candidates for an operator's definition. In the GPP process, once a candidate definition survives this empirical prune, it is passed on to a theorem prover for formal verification. We describe the application of this methodology to the (re)discovery of the definition of multiplication for Conway numbers, a discovery which is difficult for human mathematicians. We successfully model this discovery process utilizing information which was reasonably available at the time of Conway's original discovery. As part of this discovery process, we reduce the size of the search space from a computationally intractable size to 3468 elements

Spiritual sadomasochism

This thesis is an account of the author\u27s mental, physical, and artistic search for the spiritual. The author describes his search for spiritual fulfillment through sadomasochistic ritual and body piercing. This spiritual journey is portrayed through the medium of photography, displayed on 20 slides

Topological spaces associated to higher-rank graphs

We investigate which topological spaces can be constructed as topological realisations of higher-rank graphs. We describe equivalence relations on higher-rank graphs for which the quotient is again a higher-rank graph, and show that identifying isomorphic co-hereditary subgraphs in a disjoint union of two rank-$k$ graphs gives rise to pullbacks of the associated $C^*$-algebras. We describe a combinatorial version of the connected-sum operation and apply it to the rank-2-graph realisations of the four basic surfaces to deduce that every compact 2-manifold is the topological realisation of a rank-2 graph. We also show how to construct $k$-spheres and wedges of $k$-spheres as topological realisations of rank-$k$ graphs.Comment: Updated to agree with published versio

The Prophetic Imagination as the Conduit for Grace and Fertile Ground for Students on the Flatlands of a Desacralized World

Grace and the prophetic imagination are the theological wellsprings that provide the fertile ground for an encounter with God\u27s Holy Mystery, a God who transcends humanity yet who in steadfast love cares for all even to the point of becoming one of with humanity. Grace opens the hearts and minds of individuals to hear God\u27s prophetic word where God reveals God\u27s very nature and where people grow in becoming themselves fully and wholly for others. The prophetic imagination affords believers the awareness to see God\u27s incarnational love alive and present - now and everywhere. Since mystery and self-transcendence, a life and a meaning for it are foundational for a healthy religion,[1] this project will integrate various voices to formulate a much-needed pedagogy that provides a space where students are disarmed, stories shared, trust restored, and compassion resurrected thereby preparing students for an integrated life. The underpinning question this project addresses for religious education and educators in the twenty-first century is; How might we teach an alienated and increasing religiously illiterate populace, a population in Parker Palmer\u27s words, that lives on the flatlands of a desacralized world? [2] According to Palmer, the desacralized landscape is flat, tedious, and devoid of surprise or wonder. It gives no impetus to neither go beyond oneself nor explore the depths of one\u27s being, not to mention experience the mystery of God\u27s love. Transforming these flatlands is paramount and requires forming critical thinkers for the present and integrated lives for the future. [1] Thomas F. O\u27Meara, O.P., Religious Education for Maturity: The Presence of Grace, Religious Education LXVIII, no. 4 (July/August 1973): 454. [2] Parker J. Palmer, The Courage to Teach: Exploring the Inner Landscape of a Teacher\u27s Life (San Francisco, CA: Jossey-Bass, 1998) 115