On the Spectra of Real and Complex Lam\'e Operators

We study Lam\'e operators of the form $$L = -\frac{d^2}{dx^2} + m(m+1)\omega^2\wp(\omega x+z_0),$$ with $m\in\mathbb{N}$ and $\omega$ a half-period of $\wp(z)$. For rectangular period lattices, we can choose $\omega$ and $z_0$ such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lam\'e operator has a band structure with not more than $m$ gaps. In the first part of the paper, we prove that the opened gaps are precisely the first $m$ ones. In the second part, we study the Lam\'e spectrum for a generic period lattice when the potential is complex-valued. We concentrate on the $m=1$ case, when the spectrum consists of two regular analytic arcs, one of which extends to infinity, and briefly discuss the $m=2$ case, paying particular attention to the rhombic lattices

Complex exceptional orthogonal polynomials and quasi-invariance

Consider the Wronskians of the classical Hermite polynomials Hλ₁(x):= Wr(Hl(x);Hk1 (x)…;Hkn(x)); l ϵ Z≥0 \{k1; : : : ; kn}; where ki = λ₁ + n - i; i = 1;…, n and λ = (λ₁;…; λn) is a partition. Gómez-Ullate et al. showed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with respect to a certain inner product, but in contrast to the usual case have some degrees missing (so called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of C[x] satisfying certain quasi-invariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schrödinger operators, is also presented

Complex exceptional orthogonal polynomials and quasi-invariance

Consider the Wronskians of the classical Hermite polynomials Hλ₁(x):= Wr(Hl(x);Hk1 (x)…;Hkn(x)); l ϵ Z≥0 \{k1; : : : ; kn}; where ki = λ₁ + n - i; i = 1;…, n and λ = (λ₁;…; λn) is a partition. Gómez-Ullate et al. showed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with respect to a certain inner product, but in contrast to the usual case have some degrees missing (so called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of C[x] satisfying certain quasi-invariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schrödinger operators, is also presented

On the spectra of real and complex Lame operators

On the spectra of real and complex Lame operator

Clinical Judgment Versus Biomarker Prostate Cancer Gene 3: Which Is Best When Determining the Need for Repeat Prostate Biopsy?

ObjectiveTo assess the value of best clinical judgment (BCJ) and the prostate cancer gene 3 (PCA3) assay in guiding the decision to perform a repeat prostate biopsy (PBx) after a previous negative PBx.Materials and MethodsUsing the RAND/UCLA Appropriateness Method, 12 European urologists established recommendations (BCJ) for the appropriateness of PBx according to the prostate-specific antigen level, digital rectal examination findings, number of previous negative PBxs, prostate volume, and life expectancy, with and without consideration of the PCA3 scores. These recommendations were applied to 1024 subjects receiving placebo in the Reduction by Dutasteride of Prostate Cancer Events trial, including men with a previous negative PBx, a baseline prostate-specific antigen level of 2.5-10 ng/mL, and a PCA3 test performed before the protocol-mandated 2- and 4-year repeat PBxs. Three scenarios (ie, BCJ alone, BCJ with PCA3, and the PCA3 score alone) were tested for their ability to reduce the repeat PBx rate versus missing Gleason sum ≥7 prostate cancer (PCa).ResultsBCJ with PCA3 would have avoided 64% of repeat PBxs compared with 26% for BCJ alone and 55% for PCA3 alone (cutoff score 20). Of 55 PCa cases (Gleason sum ≥7), 13 would have been missed using BCJ alone compared with 7 using PCA3 (cutoff score 20) alone and 8 using BCJ plus PCA3. The diagnostic accuracy for Gleason sum ≥7 PCa of the BCJ with PCA3 scenario was superior to that of the other scenarios, with a negative predictive value of 99%.ConclusionApplication of the BCJ together with PCA3 testing can reduce the number of repeat PBxs while maintaining the sensitivity to detect Gleason sum ≥7 PCa

New strategy for the identification of prostate cancer: The combination of Proclarix and the prostate health index

Prostate health index (PHI) and, more recently, Proclarix have been proposed as serum biomarkers for prostate cancer (PCa). In this study, we aimed to evaluate Proclarix and PHI for predicting clinically significant prostate cancer (csPCa)

Accommodating heterogeneous missing data patterns for prostate cancer risk prediction

Objective: We compared six commonly used logistic regression methods for accommodating missing risk factor data from multiple heterogeneous cohorts, in which some cohorts do not collect some risk factors at all, and developed an online risk prediction tool that accommodates missing risk factors from the end-user. Study Design and Setting: Ten North American and European cohorts from the Prostate Biopsy Collaborative Group (PBCG) were used for fitting a risk prediction tool for clinically significant prostate cancer, defined as Gleason grade group greater or equal 2 on standard TRUS prostate biopsy. One large European PBCG cohort was withheld for external validation, where calibration-in-the-large (CIL), calibration curves, and area-underneath-the-receiver-operating characteristic curve (AUC) were evaluated. Ten-fold leave-one-cohort-internal validation further validated the optimal missing data approach. Results: Among 12,703 biopsies from 10 training cohorts, 3,597 (28%) had clinically significant prostate cancer, compared to 1,757 of 5,540 (32%) in the external validation cohort. In external validation, the available cases method that pooled individual patient data containing all risk factors input by an end-user had best CIL, under-predicting risks as percentages by 2.9% on average, and obtained an AUC of 75.7%. Imputation had the worst CIL (-13.3%). The available cases method was further validated as optimal in internal cross-validation and thus used for development of an online risk tool. For end-users of the risk tool, two risk factors were mandatory: serum prostate-specific antigen (PSA) and age, and ten were optional: digital rectal exam, prostate volume, prior negative biopsy, 5-alpha-reductase-inhibitor use, prior PSA screen, African ancestry, Hispanic ethnicity, first-degree prostate-, breast-, and second-degree prostate-cancer family history

Crowdsourcing Methods for Data Collection in Geophysics: State of the Art, Issues, and Future Directions

Data are essential in all areas of geophysics. They are used to better understand and manage systems, either directly or via models. Given the complexity and spatiotemporal variability of geophysical systems (e.g., precipitation), a lack of sufficient data is a perennial problem, which is exacerbated by various drivers, such as climate change and urbanization. In recent years, crowdsourcing has become increasingly prominent as a means of supplementing data obtained from more traditional sources, particularly due to its relatively low implementation cost and ability to increase the spatial and/or temporal resolution of data significantly. Given the proliferation of different crowdsourcing methods in geophysics and the promise they have shown, it is timely to assess the state‐of‐the‐art in this field, to identify potential issues and map out a way forward. In this paper, crowdsourcing‐based data acquisition methods that have been used in seven domains of geophysics, including weather, precipitation, air pollution, geography, ecology, surface water and natural hazard management are discussed based on a review of 162 papers. In addition, a novel framework for categorizing these methods is introduced and applied to the methods used in the seven domains of geophysics considered in this review. This paper also features a review of 93 papers dealing with issues that are common to data acquisition methods in different domains of geophysics, including the management of crowdsourcing projects, data quality, data processing and data privacy. In each of these areas, the current status is discussed and challenges and future directions are outlined