Generalized Ramanujan Primes

In 1845, Bertrand conjectured that for all integers $x\ge2$, there exists at least one prime in $(x/2, x]$. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any $n\ge1$, there is a (smallest) prime $R_n$ such that $\pi(x)- \pi(x/2) \ge n$ for all $x \ge R_n$. In 2009 Sondow called $R_n$ the $n$th Ramanujan prime and proved the asymptotic behavior $R_n \sim p_{2n}$ (where $p_m$ is the $m$th prime). In the present paper, we generalize the interval of interest by introducing a parameter $c \in (0,1)$ and defining the $n$th $c$-Ramanujan prime as the smallest integer $R_{c,n}$ such that for all $x\ge R_{c,n}$, there are at least $n$ primes in $(cx,x]$. Using consequences of strengthened versions of the Prime Number Theorem, we prove that $R_{c,n}$ exists for all $n$ and all $c$, that $R_{c,n} \sim p_{\frac{n}{1-c}}$ as $n\to\infty$, and that the fraction of primes which are $c$-Ramanujan converges to $1-c$. We then study finer questions related to their distribution among the primes, and see that the $c$-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping; this was first observed by Sondow, Nicholson, and Noe in the case $c = 1/2$. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales, but has been shown to fail in some instances on smaller scales.Comment: 13 pages, 2 tables, to appear in the CANT 2011 Conference Proceedings. This is version 2.0. Changes: fixed typos, added references to OEIS sequences, and cited Shevelev's preprin

Maass waveforms and low-lying zeros

The Katz-Sarnak Density Conjecture states that the behavior of zeros of a family of $L$-functions near the central point (as the conductors tend to zero) agrees with the behavior of eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo and Sarnak proved that the behavior of zeros near the central point of holomorphic cusp forms agrees with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions $\phi$. We prove similar results for families of cuspidal Maass forms, the other natural family of ${\rm GL}_2/\mathbb{Q}$ $L$-functions. For suitable weight functions on the space of Maass forms, the limiting behavior agrees with the expected orthogonal group. We prove this for \Supp(\widehat{\phi})\subseteq (-3/2, 3/2) when the level $N$ tends to infinity through the square-free numbers; if the level is fixed the support decreases to being contained in $(-1,1)$, though we still uniquely specify the symmetry type by computing the 2-level density.Comment: Version 2.1, 33 pages, expanded introduction on low-lying zeros and the Katz-Sarnak density conjecture, fixed some typo

Ultrasound shear wave elastography and liver fibrosis: A Prospective Multicenter Study

Aim: To assess the accuracy of shear wave elastography (SWE) alone and in combination with aminotransferase platelet ratio index (APRI) score in the staging of liver fibrosis. Methods: A multicenter prospective study was conducted to assess the accuracy of SWE (medians) and APRI to predict biopsy results. The analysis focused on distinguishing the different stages of liver disease, namely, F0 from F1-4, F0-1 from F2-4, F0-2 from F3-4 and F0-3 from F4; F0-F1 from F2-F4 being of primary interest. The area under the receiver operating characteristic (AUROC) curve was computed using logistic regression model. The role of age, gender and steatosis was also assessed. Results: SWE alone accurately distinguished F0-1 from F2-4 with a high probability. The AUROC using SWE alone was 0.91 compared to 0.78 for using the APRI score alone. The APRI score, when used in conjunction with SWE, did not make a significant contribution to the AUROC. SWE and steatosis were the only significant predictors that differentiated F0-1 from F2-4 with an AUROC of 0.944. Conclusion: Our study validates the use of SWE in the diagnosis and staging of liver fibrosis. Furthermore, the probability of a correct diagnosis is significantly enhanced with the addition of steatosis as a prognostic factor

Translational Challenges and Therapeutic Opportunities in BRCA1-Related Breast Cancer

Although significant progress has been made in the management of the hereditary cancer syndrome related to mutations of BRCA1, two fundamental and clinically relevant questions regarding BRCA1-related cancer syndrome remain unresolved: (1) What factors account for the tissue specificity of the BRCA1-related cancer risk? (2) How does a mutation or loss of BRCA1 lead to the basal-like phenotype of breast cancer? This review focuses on recent studies in BRCA1-related pathways that lead to specific characteristics of the hereditary cancer syndrome and discusses the current translational evidence for exploiting these pathways in new therapeutic strategies. Mounting evidence suggests that estrogen signaling and metabolism, oxidative stress, specific secondary mutations, and regulation of specific progenitor cells and transcriptional programs are critical in BRCA1-associated breast cancer. Strategies geared toward estrogen reduction may play a role in treatment and prevention. Therapies aimed at mitigating oxidative stress may be a strategy for risk reduction, while cancer-cell-specific sensitivity to oxidative stress may also be an opportunity for specific targeting. BRCA1-related transcriptional regulation and signaling provide a number of therapeutic targets, including the PI3-AKT and Notch pathways. Thus, significant opportunities exist in translational and clinical research for developing the treatment strategies for the management of BRCA1-related breast cancer

Promoting clinical breast evaluations in a lower middle–income country Setting: An approach toward achieving a sustainable breast health program

PURPOSE: To promote a systems-based approach for the early detection and downstaging of breast cancer at presentation in the remote mountainous region of Gorno Badakhshan Autonomous Oblast (GBAO), Tajikistan, by introducing breast cancer awareness into the community and training health care professionals in clinical breast evaluation (CBE). METHODS: Through a public-private partnership between the Ministry of Health, the Aga Khan Health Services, Tajikistan, and the Aga Khan Health Board, we organized breast cancer screening in the community and trained family medicine doctors (FMDs) and family medicine nurses (FMNs) in CBE. We identified and trained CBE master trainers, who, in turn, systematically trained FMNs to conduct CBEs in each of the remote regional clinics. RESULTS: Between 2014 and 2017, 47 FMDs (85% of all FMDs in GBAO), 166 FMNs (55% of all FMNs in GBAO), and six master trainers were trained. Of 3,556 women who were screened, abnormal CBEs were noted in 696 of them (20%). Of the last 1,101 CBEs that were performed by trainee FMNs, with secondary CBEs by master trainers, the rate of abnormal CBEs plateaued at 9%. A total of 18 women were diagnosed with breast cancer-2.6% of abnormal CBEs and 0.5% of all screened women. CONCLUSION: A dual-pronged approach of community awareness and CBE training of health care professionals, supported by CBE master trainers, offers a sustainable approach for the early detection of breast pathology. We observed anecdotal evidence of clinical early-stage detection over time with improved CBE proficiency and community acceptance. Sustaining this program will require advocacy by health care providers and a responsive public policy to support the early detection and treatment of breast cancer across the region

Inference for Constrained Estimation of Tumor Size Distributions

In order to develop better treatment and screening programs for cancer prevention programs, it is important to be able to understand the natural history of the disease and what factors affect its progression. We focus on a particular framework first outlined by Kimmel and Flehinger (1991, Biometrics , 47, 987–1004) and in particular one of their limiting scenarios for analysis. Using an equivalence with a binary regression model, we characterize the nonparametric maximum likelihood estimation procedure for estimation of the tumor size distribution function and give associated asymptotic results. Extensions to semiparametric models and missing data are also described. Application to data from two cancer studies is used to illustrate the finite-sample behavior of the procedure.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/65536/1/j.1541-0420.2008.01001.x.pd

Brazilian gamma detection device for sentinel lymph node biopsy

OBJECTIVE: To test the effectiveness of an intra-operative gamma detection Brazilian device (IPEN) on sentinel lymph node biopsy (SLNB) procedures. METHODS: Forty melanoma or breast cancer patients with indication for undergoing SLNB were studied. Lymphoscintigraphy was done 2 to 24 hours prior to surgery. Lymphatic mapping with vital dye and gamma detection were performed intraoperatively. For gamma detection Neoprobe ® 1500 was used followed by IPEN (equipment under test) in the first 20 patients and for the remaining half IPEN was used first to verify its ability to locate the sentinel node (SN). Measurements were taken from the radiopharmaceutical product injection site, from SN (in vivo and ex vivo) and from background. It was recorded if the SN was stained or not and if it was found easily by surgeon. RESULTS: There were 33 (82.5%) breast cancer and 7 (17.5%) melanoma patients. Ages varied from 21 to 68 year-old (median age of 46). Sex distribution was 35 (87.5%) women and 5 (12.5%) men. Sentinel node was found in all but one patient. There was no statistical difference between the reasons ex vivo/ background obtained with the measures of both equipments (p=0, 2583-ns). The SN was easily found by the surgeon with both devices. CONCLUSION: The SLNB was successfully performed using either equipment. It was possible to do SLNB with the Brazilian device developed by IPEN without prejudice for the patient.OBJETIVO: Testar a eficácia de equipamento de detecção gama intra-operatória (DGI) desenvolvido pelo IPEN (Brasil), em procedimentos de biópsia de linfonodo sentinela (BLS) no melanoma e no câncer de mama. MÉTODOS: Foram estudados 40 pacientes portadores de melanoma ou câncer de mama com indicação para realização de BLS.Todos pacientes foram submetidos à linfocintilografia e a BLS ocorreu entre 2 a 24 horas após a mesma. Concomitantemente à DGI, realizou-se o mapeamento linfático com corante vital. Foram feitas leituras com o equipamento convencional Neoprobe® 1500 e com o equipamento em teste (IPEN) dos valores de captação do sítio de injeção do radiofármaco, do LS in vivo e ex vivo e da captação de fundo. Foi registrado se o LS estava corado e se o cirurgião teve facilidade para encontrá-lo. Nos primeiros 20 pacientes utilizou-se o equipamento convencional e depois o de teste; nos outros 20, utilizou-se primeiro o equipamento em teste, com objetivo de verificar se o mesmo identificava primariamente o LS. RESULTADOS: Dos quarenta pacientes, 33 eram portadores de tumor de mama e sete de melanoma cutâneo; variação da idade: 21 a 68 anos (mediana= 46 anos); 35 mulheres e 5 homens. Em apenas um paciente o LS não foi encontrado, nem pela DGI nem pelo corante vital. Não houve diferença estatística entre as razões ex vivo/fundo obtidas com os dois equipamentos (p=0, 2583-ns). CONCLUSÃO: É possível realizar o procedimento de BLS com o equipamento brasileiro desenvolvido pelo IPEN, com facilidade e sem prejuízo para o paciente.Hospital Israelita Albert EinsteinIPENUniversidade Federal de São Paulo (UNIFESP)Faculdade de Medicina Anhembi-MorumbiUNIFESPSciEL