Elementary Differential Equations with Boundary Value Problems

Written in a clear and accurate language that students can understand, Trench\u27s new book minimizes the number of explicitly stated theorems and definitions. Instead, he deals with concepts in a conversational style that engages students. He includes more than 250 illustrated, worked examples for easy reading and comprehension. One of the book\u27s many strengths is its problems, which are of consistently high quality. Trench includes a thorough treatment of boundary-value problems and partial differential equations and has organized the book to allow instructors to select the level of technology desired. This has been simplified by using symbols, C and L, to designate the level of technology. C problems call for computations and/or graphics, while L problems are laboratory exercises that require extensive use of technology. Informal advice on the use of technology is included in several sections and instructors who prefer not to emphasize technology can ignore these exercises without interrupting the flow of material. (From the 1st edition)https://digitalcommons.trinity.edu/mono/1008/thumbnail.jp

On the eigenvalue problem for Toeplitz band matrices

AbstractA formula is given for the characteristic polynomial of an nth order Toeplitz band matrix, with bandwidth k < n, in terms of the zeros of a kth degree polynomial with coefficients independent of n. The complexity of the formula depends on the bandwidth k, and not on the order n. Also given is a formula for eigenvectors, in terms of the same zeros and k coefficients which can be obtained by solving a k × k homogeneous system

Introduction to Real Analysis

Using a clear and informal approach, this book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.https://digitalcommons.trinity.edu/mono/1006/thumbnail.jp

Student Solutions Manual for Elementary Differential Equations and Elementary Differential Equations with Boundary Value Problems

This book was published previously by Brooks/Cole Thomson Learning. This book has been judged to meet the evaluation criteria set by the Editorial Board of theAmerican Institute ofMathematics in connection with the Institute’s Open Textbook Initiative. It may be copied, modified, redistributed, translated, and built upon subject to the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.https://digitalcommons.trinity.edu/mono/1009/thumbnail.jp

Evaluation of polygenic risk scores for breast and ovarian cancer risk prediction in BRCA1 and BRCA2 mutation carriers

Background: Genome-wide association studies (GWAS) have identified 94 common single-nucleotide polymorphisms (SNPs) associated with breast cancer (BC) risk and 18 associated with ovarian cancer (OC) risk. Several of these are also associated with risk of BC or OC for women who carry a pathogenic mutation in the high-risk BC and OC genes BRCA1 or BRCA2. The combined effects of these variants on BC or OC risk for BRCA1 and BRCA2 mutation carriers have not yet been assessed while their clinical management could benefit from improved personalized risk estimates. Methods: We constructed polygenic risk scores (PRS) using BC and OC susceptibility SNPs identified through population-based GWAS: for BC (overall, estrogen receptor [ER]-positive, and ER-negative) and for OC. Using data from 15 252 female BRCA1 and 8211 BRCA2 carriers, the association of each PRS with BC or OC risk was evaluated using a weighted cohort approach, with time to diagnosis as the outcome and estimation of the hazard ratios (HRs) per standard deviation increase in the PRS. Results: The PRS for ER-negative BC displayed the strongest association with BC risk in BRCA1 carriers (HR = 1.27, 95% confidence interval [CI] = 1.23 to 1.31, P = 8.2 x 10(53)). In BRCA2 carriers, the strongest association with BC risk was seen for the overall BC PRS (HR = 1.22, 95% CI = 1.17 to 1.28, P = 7.2 x 10(-20)). The OC PRS was strongly associated with OC risk for both BRCA1 and BRCA2 carriers. These translate to differences in absolute risks (more than 10% in each case) between the top and bottom deciles of the PRS distribution; for example, the OC risk was 6% by age 80 years for BRCA2 carriers at the 10th percentile of the OC PRS compared with 19% risk for those at the 90th percentile of PRS. Conclusions: BC and OC PRS are predictive of cancer risk in BRCA1 and BRCA2 carriers. Incorporation of the PRS into risk prediction models has promise to better inform decisions on cancer risk management

BRCA1 and BRCA2 pathogenic sequence variants in women of African origin or ancestry.

BRCA1 and BRCA2 (BRCA1/2) pathogenic sequence variants (PSVs) confer elevated risks of multiple cancers. However, most BRCA1/2 PSVs reports focus on European ancestry individuals. Knowledge of the PSV distribution in African descent individuals is poorly understood. We undertook a systematic review of the published literature and publicly available databases reporting BRCA1/2 PSVs also accessed the Consortium of Investigators of Modifiers of BRCA1/2 (CIMBA) database to identify African or African descent individuals. Using these data, we inferred which of the BRCA PSVs were likely to be of African continental origin. Of the 43,817 BRCA1/2 PSV carriers in the CIMBA database, 469 (1%) were of African descent. Additional African descent individuals were identified in public databases (n = 291) and the literature (n = 601). We identified 164 unique BRCA1 and 173 unique BRCA2 PSVs in individuals of African ancestry. Of these, 83 BRCA1 and 91 BRCA2 PSVs are of likely or possible African origin. We observed numerous differences in the distribution of PSV type and function in African origin versus non-African origin PSVs. Research in populations of African ancestry with BRCA1/2 PSVs is needed to provide the information needed for clinical management and decision-making in African descent individuals worldwide

The SNP rs6500843 in 16p13.3 is associated with survival specifically among chemotherapy-treated breast cancer patients

We have utilized a two-stage study design to search for SNPs associated with the survival of breast cancer patients treated with adjuvant chemotherapy. Our initial GWS data set consisted of 805 Finnish breast cancer cases (360 treated with adjuvant chemotherapy). The top 39 SNPs from this stage were analyzed in three independent data sets: iCOGS (n=6720 chemotherapy-treated cases), SUCCESS-A (n=3596), and POSH (n=518). Two SNPs were successfully validated: rs6500843 (any chemotherapy; per-allele HR 1.16, 95% C.I. 1.08-1.26, p=0.0001, p((adjusted))=0.0091), and rs11155012 (anthracycline therapy; per-allele HR 1.21, 95% C.I. 1.08-1.35, p=0.0010, p((adjusted))=0.0270). The SNP rs6500843 was found to specifically interact with adjuvant chemotherapy, independently of standard prognostic markers (p((interaction))=0.0009), with the rs6500843-GG genotype corresponding to the highest hazard among chemotherapy-treated cases (HR 1.47, 95% C.I. 1.20-1.80). Upon trans-eQTL analysis of public microarray data, the rs6500843 locus was found to associate with the expression of a group of genes involved in cell cycle control, notably AURKA, the expression of which also exhibited differential prognostic value between chemotherapy-treated and untreated cases in our analysis of microarray data. Based on previously published information, we propose that the eQTL genes may be connected to the rs6500843 locus via a RBFOX1-FOXM1 -mediated regulatory pathway.Peer reviewe

Ovarian and Breast Cancer Risks Associated With Pathogenic Variants in RAD51C and RAD51D.

BACKGROUND: The purpose of this study was to estimate precise age-specific tubo-ovarian carcinoma (TOC) and breast cancer (BC) risks for carriers of pathogenic variants in RAD51C and RAD51D. METHODS: We analyzed data from 6178 families, 125 with pathogenic variants in RAD51C, and 6690 families, 60 with pathogenic variants in RAD51D. TOC and BC relative and cumulative risks were estimated using complex segregation analysis to model the cancer inheritance patterns in families while adjusting for the mode of ascertainment of each family. All statistical tests were two-sided. RESULTS: Pathogenic variants in both RAD51C and RAD51D were associated with TOC (RAD51C: relative risk [RR] = 7.55, 95% confidence interval [CI] = 5.60 to 10.19; P = 5 × 10-40; RAD51D: RR = 7.60, 95% CI = 5.61 to 10.30; P = 5 × 10-39) and BC (RAD51C: RR = 1.99, 95% CI = 1.39 to 2.85; P = 1.55 × 10-4; RAD51D: RR = 1.83, 95% CI = 1.24 to 2.72; P = .002). For both RAD51C and RAD51D, there was a suggestion that the TOC relative risks increased with age until around age 60 years and decreased thereafter. The estimated cumulative risks of developing TOC to age 80 years were 11% (95% CI = 6% to 21%) for RAD51C and 13% (95% CI = 7% to 23%) for RAD51D pathogenic variant carriers. The estimated cumulative risks of developing BC to 80 years were 21% (95% CI = 15% to 29%) for RAD51C and 20% (95% CI = 14% to 28%) for RAD51D pathogenic variant carriers. Both TOC and BC risks for RAD51C and RAD51D pathogenic variant carriers varied by cancer family history and could be as high as 32-36% for TOC, for carriers with two first-degree relatives diagnosed with TOC, or 44-46% for BC, for carriers with two first-degree relatives diagnosed with BC. CONCLUSIONS: These estimates will facilitate the genetic counseling of RAD51C and RAD51D pathogenic variant carriers and justify the incorporation of RAD51C and RAD51D into cancer risk prediction models