On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions

Given a finite simple graph \cG with $n$ vertices, we can construct the Cayley graph on the symmetric group $S_n$ generated by the edges of \cG, interpreted as transpositions. We show that, if \cG is complete multipartite, the eigenvalues of the Laplacian of \Cay(\cG) have a simple expression in terms of the irreducible characters of transpositions, and of the Littlewood-Richardson coefficients. As a consequence we can prove that the Laplacians of \cG and of \Cay(\cG) have the same first nontrivial eigenvalue. This is equivalent to saying that Aldous's conjecture, asserting that the random walk and the interchange process have the same spectral gap, holds for complete multipartite graphs.Comment: 29 pages. Includes modification which appear on the published version in J. Algebraic Combi

A Retrospective Analysis of Opioid Consumption Among Different Orthopedic Surgeons for Total Joint Replacement

Background: Throughout the world, baby boomers reaching their sixth, seventh, and eighth decade of life are requiring a significant number of joint replacements—hips and knees. Due to the increasing number of joint replacements, it is important to find a multi-modal approach (MMA) to control pain, reduce the amount of opioid consumption, and improve patient satisfaction. Purpose: The purpose of this study was to evaluate the intraoperative, postoperative, and total opioid consumption of patients undergoing total hip and knee replacements in an effort to develop a multi-modal approach to decrease opioid consumption, minimize adverse effects secondary to narcotic administration, and to achieve better pain control. This MMA was achieved by administering oxycodone, gabapentin, celecoxib, and acetaminophen starting before surgical incision. Methods: The study sample consisted of 192 patients undergoing total hip and knee replacements over a 10-month period between June 2012 and March 2013 at UMASS Memorial performed by five orthopedic surgeons. The main objective was to record intraoperative, postoperative, total opioid consumption, and patient satisfaction amongst these patients. Furthermore, the patients were subdivided based on the type of procedure (hip vs knee), type of anesthetic (general vs spinal), and the presence or absence of an indwelling catheter to deliver anesthetic (catheter vs no catheter). Results: The data showed a large variability among the surgeons in regards to the amount of opioid used intraoperatively, postoperatively and total opioid consumption. In terms of type of anesthetic, the patients undergoing spinal anesthesia used statistically significantly less opioids intraoperatively but not postoperatively, compared to general anesthesia. As for catheter use with general and spinal anesthesia, surprisingly, there was no significant difference in opioid consumption compared to the non-catheter counterpart. Furthermore, there seems to be no correlation between body mass index (BMI) and intraoperative or postoperative opioid use. Patient satisfaction was another variable that showed no correlation with opioid use intraoperatively or postoperatively. In terms of age, the data suggests that older patients use less opioids postoperatively in both hip and knee replacements. Conclusions: Our results quantitatively show spinal anesthesia to be far superior than general anesthesia in both joint replacements. Spinal anesthesia provides better pain control intraoperatively which allows one to use less opioids, thereby minimizing the adverse side effects of narcotic administration which include respiratory depression, urinary retention, nausea and post-operative ileus to name just a few. One surgeon’s patients required significantly less opioids intraoperatively compared to the rest of the surgeons. Further studies might warrant examining this surgeon’s technique or the demographics of his patient population to determine how better pain control and less opioid consumption could be achieved across all joints with all participating surgeons

Asymptotic Expansions for the Conditional Sojourn Time Distribution in the M/M/1M/M/1-PS Queue

We consider the $M/M/1$ queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer's service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison \cite{MO} and Flatto \cite{FL}.Comment: 30 pages, 3 figures and 1 tabl

Ordering of Energy Levels in Heisenberg Models and Applications

In a recent paper we conjectured that for ferromagnetic Heisenberg models the smallest eigenvalues in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin and called this property ferromagnetic ordering of energy levels (FOEL). We have proved this conjecture for the Heisenberg model with arbitrary spins and coupling constants on a chain. In this paper we give a pedagogical introduction to this result and also discuss some extensions and implications. The latter include the property that the relaxation time of symmetric simple exclusion processes on a graph for which FOEL can be proved, equals the relaxation time of a random walk on the same graph. This equality of relaxation times is known as Aldous' Conjecture.Comment: 20 pages, contribution for the proceedings of QMATH9, Giens, September 200

On the injectivity of the circular Radon transform arising in thermoacoustic tomography

The circular Radon transform integrates a function over the set of all spheres with a given set of centers. The problem of injectivity of this transform (as well as inversion formulas, range descriptions, etc.) arises in many fields from approximation theory to integral geometry, to inverse problems for PDEs, and recently to newly developing types of tomography. The article discusses known and provides new results that one can obtain by methods that essentially involve only the finite speed of propagation and domain dependence for the wave equation.Comment: To appear in Inverse Problem

Random division of an interval

The well-known relation between random division of an interval and the Poisson process is interpreted as a Laplace transformation. With the use of this interpretation a number of (in part known) results is derived very easily

Level Sets of the Takagi Function: Local Level Sets

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a "generic" full Lebesgue measure set of ordinates y, the level sets are finite sets. Here it is shown for a "generic" full Lebesgue measure set of abscissas x, the level set L(\tau(x)) is uncountable. An interesting singular monotone function is constructed, associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly 3/2.Comment: 32 pages, 2 figures, 1 table. Latest version has updated equation numbering. The final publication will soon be available at springerlink.co

Harmonic analysis of iterated function systems with overlap

In this paper we extend previous work on IFSs without overlap. Our method involves systems of operators generalizing the more familiar Cuntz relations from operator algebra theory, and from subband filter operators in signal processing.Comment: 37 page