Divisible Codes - A Survey

This paper surveys parts of the study of divisibility properties of codes. The survey begins with the motivating background involving polynomials over finite fields. Then it presents recent results on bounds and applications to optimal codes.

The Value Of The 2005 International Society Of Urological Pathology (isup) Modified Gleason Grading System As A Predictor Of Biochemical Recurrence After Radical Prostatectomy.

To compare time and risk to biochemical recurrence (BR) after radical prostatectomy of two chronologically different groups of patients using the standard and the modified Gleason system (MGS). Cohort 1 comprised biopsies of 197 patients graded according to the standard Gleason system (SGS) in the period 1997/2004, and cohort 2, 176 biopsies graded according to the modified system in the period 2005/2011. Time to BR was analyzed with the Kaplan-Meier product-limit analysis and prediction of shorter time to recurrence using univariate and multivariate Cox proportional hazards model. Patients in cohort 2 reflected time-related changes: striking increase in clinical stage T1c, systematic use of extended biopsies, and lower percentage of total length of cancer in millimeter in all cores. The MGS used in cohort 2 showed fewer biopsies with Gleason score ≤ 6 and more biopsies of the intermediate Gleason score 7. Time to BR using the Kaplan-Meier curves showed statistical significance using the MGS in cohort 2, but not the SGS in cohort 1. Only the MGS predicted shorter time to BR on univariate analysis and on multivariate analysis was an independent predictor. The results favor that the 2005 International Society of Urological Pathology modified system is a refinement of the Gleason grading and valuable for contemporary clinical practice.

Gene expression relationship between prostate cancer cells of Gleason 3, 4 and normal epithelial cells as revealed by cell type-specific transcriptomes

Background: Prostate cancer cells in primary tumors have been typed CD10(-)/CD13(-)/CD24(hi)/CD26(+)/CD38(lo)/CD44(-)/CD104(-). This CD phenotype suggests a lineage relationship between cancer cells and luminal cells. The Gleason grade of tumors is a descriptive of tumor glandular differentiation. Higher Gleason scores are associated with treatment failure. Methods: CD26(+) cancer cells were isolated from Gleason 3+3 (G3) and Gleason 4+4 (G4) tumors by cell sorting, and their gene expression or transcriptome was determined by Affymetrix DNA array analysis. Dataset analysis was used to determine gene expression similarities and differences between G3 and G4 as well as to prostate cancer cell lines and histologically normal prostate luminal cells. Results: The G3 and G4 transcriptomes were compared to those of prostatic cell types of non-cancer, which included luminal, basal, stromal fibromuscular, and endothelial. A principal components analysis of the various transcriptome datasets indicated a closer relationship between luminal and G3 than luminal and G4. Dataset comparison also showed that the cancer transcriptomes differed substantially from those of prostate cancer cell lines. Conclusions: Genes differentially expressed in cancer are potential biomarkers for cancer detection, and those differentially expressed between G3 and G4 are potential biomarkers for disease stratification given that G4 cancer is associated with poor outcomes. Differentially expressed genes likely contribute to the prostate cancer phenotype and constitute the signatures of these particular cancer cell types.

A Goldstone theorem in thermal relativistic quantum field theory

We prove a Goldstone theorem in thermal relativistic quantum field theory, which relates spontaneous symmetry breaking to the rate of spacelike decay of the two-point function. The critical rate of fall-off coincides with that of the massless free scalar field theory. Related results and open problems are briefly discussed. (C) 2011 American Institute of Physics. [doi:10.1063/1.3526961]

An algorithmic Friedman-Pippenger theorem on tree embeddings and applications

An (n, d)-expander is a graph G = (V, E) such that for every X subset of V with vertical bar X vertical bar <= 2n - 2 we have vertical bar Gamma(G)(X) vertical bar >= (d + 1) vertical bar X vertical bar. A tree T is small if it has at most n vertices and has maximum degree at most d. Friedman and Pippenger (1987) proved that any ( n; d)- expander contains every small tree. However, their elegant proof does not seem to yield an efficient algorithm for obtaining the tree. In this paper, we give an alternative result that does admit a polynomial time algorithm for finding the immersion of any small tree in subgraphs G of (N, D, lambda)-graphs Lambda, as long as G contains a positive fraction of the edges of Lambda and lambda/D is small enough. In several applications of the Friedman-Pippenger theorem, including the ones in the original paper of those authors, the (n, d)-expander G is a subgraph of an (N, D, lambda)-graph as above. Therefore, our result suffices to provide efficient algorithms for such previously non-constructive applications. As an example, we discuss a recent result of Alon, Krivelevich, and Sudakov (2007) concerning embedding nearly spanning bounded degree trees, the proof of which makes use of the Friedman-Pippenger theorem. We shall also show a construction inspired on Wigderson-Zuckerman expander graphs for which any sufficiently dense subgraph contains all trees of sizes and maximum degrees achieving essentially optimal parameters. Our algorithmic approach is based on a reduction of the tree embedding problem to a certain on-line matching problem for bipartite graphs, solved by Aggarwal et al. (1996).

A NEW PROOF OF OKAJI'S THEOREM FOR A CLASS OF SUM OF SQUARES OPERATORS

Let P be a linear partial differential operator with analytic coefficients. We assume that P is of the form ""sum of squares"", satisfying Hormander's bracket condition. Let q be a characteristic point; for P. We assume that q lies on a symplectic Poisson stratum of codimension two. General results of Okaji Show that P is analytic hypoelliptic at q. Hence Okaji has established the validity of Treves' conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of this fact.

On the Jager-Kaul theorem concerning harmonic maps

In 1983, Jager and Kaul proved that the equator map u*(x) = (x/\x\,0) : B-n --> S-n is unstable for 3 less than or equal to n less than or equal to 6 and a minimizer for the energy functional E(u, B-n) = integral B-n \del u\(2) dx in the class H-1,H-2(B-n, S-n) with u = u* on partial derivative B-n when n greater than or equal to 7. In this paper, we give a new and elementary proof of this Jager-Kaul result. We also generalize the Jager-Kaul result to the case of p-harmonic maps.

Exponential Rates of Convergence in the Ergodic Theorem: A Constructive Approach

We prove that, once an algorithm of perfect simulation for a stationary and ergodic random field F taking values in S(Zd), S a bounded subset of R(n), is provided, the speed of convergence in the mean ergodic theorem occurs exponentially fast for F. Applications from (non-equilibrium) statistical mechanics and interacting particle systems are presented.

UPGRADING THE GLEASON SCORE IN EXTENDED PROSTATE BIOPSY: IMPLICATIONS FOR TREATMENT CHOICE

Purpose: To determine the incidence of overestimation of Gleason score (GS) in extended prostate biopsy, and consequently circumventing unnecessary aggressive treatment. Methods and Materials: This is a retrospective study of 464 patients who underwent prostate biopsy and radical prostatectomy between January 2001 and November 2007. The GS from biopsy and radical prostatectomy were compared. The incidence of overestimation of GS in biopsies and tumor volume were studied. Multivariate analysis was applied to find parameters that predict upgrading the GS in prostate biopsy. Results: The exact agreement of GS between prostate biopsy and radical prostatectomy occurred in 56.9% of cases. In 29.1% cases it was underestimated, and it was overestimated in 14%. One hundred and six (22.8%) patients received a diagnosis of high GS (8, 9, or 10) in a prostate biopsy. In 29.2% of cases, the definitive Gleason Score was 7 or lower. In cases in which GS was overestimated in the biopsy, tumors were significantly smaller. In multivariate analysis, the total percentage of tumor was the only independent factor in overestimation of GS. Tumors occupying less than 33% of cores had a 5.6-fold greater chance of being overestimated. Conclusion: In the extended biopsy era and after the International Society of Urological Pathology consensus on G, almost one third of tumors considered to have high GS at the biopsy may be intermediate-risk cancers. In that condition, tumors are smaller in biopsy. This should be remembered by professionals involved with prostate cancer to avoid overtreatment and undesirable side effects. (c) 2009 Elsevier Inc.

The effect of the number of biopsy cores on the concordance between prostate biopsy and prostatectomy Gleason score - A prostate volume-controlled study

Context.-Studies analyzing the concordance of biopsy and radical prostatectomy (RP) Gleason scores have limitations. Some included 2 or more centers, used historical controls from the early prostate specific antigen era or lacked a clear definition of the biopsy schemes. Furthermore, most did not control the results for prostate volume. Objective.-To confirm whether prediction of RP Gleason score can be optimized by taking more biopsy cores in a contemporary series of patients, with pathologic samples analyzed by the same pathologist, and controlling these results for prostate volume. Design.-The study comprised a retrospective case-control analysis of 393 patients with prostate cancer treated with RP. Patients were divided into 3 groups: those in group 1 underwent a 6-core biopsy; group 2, an 8-core biopsy; and group 3, a 10 or more-core biopsy. Concordance rates between biopsy and RP Gleason scores, as well as the rates of undergrading and overgrading, were determined for each biopsy scheme. Results.-Concordance rates were 60.9%, 58.3%, and 64.6% for patients from groups 1, 2, and 3, respectively (P = .18). When we analyzed patients with prostate volumes of less than 50 cm(3), concordance rates were 58.3%, 58.3%, and 65.1% for each group, respectively (P = .03). Among patients with prostate volumes of 50 cm3 or more, concordance rates were 70%, 58.1%, and 63.6%, respectively (P = .66). Conclusions.-Taking 10 or more cores can improve the prediction of RP Gleason score in patients with prostate volumes of less than 50 cm3. For patients with prostate volumes of 50 cm3 or more, increasing the biopsy cores to 10 or more did not improve prediction of RP Gleason score.

A fuzzy max-flow min-cut theorem

Interval-valued versions of the max-flow min-cut theorem and Karp-Edmonds algorithm are developed and provide robustness estimates for flows in networks in an imprecise or uncertain environment. These results are extended to networks with fuzzy capacities and flows. (C) 2001 Elsevier Science B.V. All rights reserved.

A flux ratio theorem for multicomponent linear diffusion equations

Ussing [1] considered the steady flux of a single chemical component diffusing through a membrane under the influence of chemical potentials and derived from his linear model, an expression for the ratio of this flux and that of the complementary experiment in which the boundary conditions were interchanged. Here, an extension of Ussing's flux ratio theorem is obtained for n chemically interacting components governed by a linear system of diffusion-migration equations that may also incorporate linear temporary trapping reactions. The determinants of the output flux matrices for complementary experiments are shown to satisfy an Ussing flux ratio formula for steady state conditions of the same form as for the well-known one-component case. (C) 2000 Elsevier Science Ltd. All rights reserved.

On a theorem of Cooper

In this paper we study some purely mathematical considerations that arise in a paper of Cooper on the foundations of thermodynamics that was published in this journal. Connections with mathematical utility theory are studied and some errors in Cooper's paper are rectified. (C) 2001 Academic Press.

Thue's theorem and the diophantine equation x2 - Dy2 = ±N

A constructive version of a theorem of Thue is used to provide representations of certain integers as x(2) - Dy-2, where D = 2, 3, 5, 6, 7.

Stereological and other methods applied to rock joint size estimation - Does Crofton's theorem apply?

A number of authors concerned with the analysis of rock jointing have used the idea that the joint areal or diametral distribution can be linked to the trace length distribution through a theorem attributed to Crofton. This brief paper seeks to demonstrate why Crofton's theorem need not be used to link moments of the trace length distribution captured by scan line or areal mapping to the moments of the diametral distribution of joints represented as disks and that it is incorrect to do so. The valid relationships for areal or scan line mapping between all the moments of the trace length distribution and those of the joint size distribution for joints modeled as disks are recalled and compared with those that might be applied were Crofton's theorem assumed to apply. For areal mapping, the relationship is fortuitously correct but incorrect for scan line mapping.