Recurrent proofs of the irrationality of certain trigonometric values

We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary transcendental functions are also discussed

Mean Value Theorems for Analytic Functions

Questions related to the location of zeros and critical points of classes of functions (polynomial, entire, analytic in a certain domain, etc.) are fundamentally important in Analysis. In this talk, he will examine some interesting mean value theorems concerning real and complex analytic functions, focusing on the complex case. He will also present sharper versions of two known results. Part of the presentation will pay tribute to the remarkable contributions of several classical Bulgarian mathematicians to problems involving the distribution of zeros of a function and its derivative(s)

IN EULER’S FOOTSTEPS: THE ENDURING APPEAL OF SPECIAL FUNCTIONS AND SPECIAL PROBLEMS

We denote the Euler-Riemann zeta function by ζ(x) and the dilogarithm by (x). The question of determining the exact value of ζ(2) (known as the Basel Problem), the one of obtaining as much information as possible about ζ(3), and a host of other related problems have been of unwavering interest for over 300 years. Several other special functions arise from the consideration of series similar to (x). Two of them are Ramanujan\u27s inverse tangent integral and Legendre\u27s chi-function . In our talk we shall derive the power series expansion for the function and use it to obtain several rapidly convergent numerical series involving zeta values. An integral representation for ζ(2) similar to the one given by Margrethe Munthe Hjortnaes in 1953 for ζ(3) is also obtained, as well as a one-line solution to the Basel problem famously settled by Euler in 1734

Mean Value Theorems for Analytic Functions

[Марков Любомир; Markov Lubomir]We prove a sharper Evard-Jafari Theorem, various mean value theorems, and an improved version of the Davitt-Powers-Riedel-Sahoo Theorem. 2010 Mathematics Subject Classification: 30C15

Ticagrelor in patients with diabetes and stable coronary artery disease with a history of previous percutaneous coronary intervention (THEMIS-PCI) : a phase 3, placebo-controlled, randomised trial

Background: Patients with stable coronary artery disease and diabetes with previous percutaneous coronary intervention (PCI), particularly those with previous stenting, are at high risk of ischaemic events. These patients are generally treated with aspirin. In this trial, we aimed to investigate if these patients would benefit from treatment with aspirin plus ticagrelor. Methods: The Effect of Ticagrelor on Health Outcomes in diabEtes Mellitus patients Intervention Study (THEMIS) was a phase 3 randomised, double-blinded, placebo-controlled trial, done in 1315 sites in 42 countries. Patients were eligible if 50 years or older, with type 2 diabetes, receiving anti-hyperglycaemic drugs for at least 6 months, with stable coronary artery disease, and one of three other mutually non-exclusive criteria: a history of previous PCI or of coronary artery bypass grafting, or documentation of angiographic stenosis of 50% or more in at least one coronary artery. Eligible patients were randomly assigned (1:1) to either ticagrelor or placebo, by use of an interactive voice-response or web-response system. The THEMIS-PCI trial comprised a prespecified subgroup of patients with previous PCI. The primary efficacy outcome was a composite of cardiovascular death, myocardial infarction, or stroke (measured in the intention-to-treat population). Findings: Between Feb 17, 2014, and May 24, 2016, 11 154 patients (58% of the overall THEMIS trial) with a history of previous PCI were enrolled in the THEMIS-PCI trial. Median follow-up was 3·3 years (IQR 2·8–3·8). In the previous PCI group, fewer patients receiving ticagrelor had a primary efficacy outcome event than in the placebo group (404 [7·3%] of 5558 vs 480 [8·6%] of 5596; HR 0·85 [95% CI 0·74–0·97], p=0·013). The same effect was not observed in patients without PCI (p=0·76, p interaction=0·16). The proportion of patients with cardiovascular death was similar in both treatment groups (174 [3·1%] with ticagrelor vs 183 (3·3%) with placebo; HR 0·96 [95% CI 0·78–1·18], p=0·68), as well as all-cause death (282 [5·1%] vs 323 [5·8%]; 0·88 [0·75–1·03], p=0·11). TIMI major bleeding occurred in 111 (2·0%) of 5536 patients receiving ticagrelor and 62 (1·1%) of 5564 patients receiving placebo (HR 2·03 [95% CI 1·48–2·76], p<0·0001), and fatal bleeding in 6 (0·1%) of 5536 patients with ticagrelor and 6 (0·1%) of 5564 with placebo (1·13 [0·36–3·50], p=0·83). Intracranial haemorrhage occurred in 33 (0·6%) and 31 (0·6%) patients (1·21 [0·74–1·97], p=0·45). Ticagrelor improved net clinical benefit: 519/5558 (9·3%) versus 617/5596 (11·0%), HR=0·85, 95% CI 0·75–0·95, p=0·005, in contrast to patients without PCI where it did not, p interaction=0·012. Benefit was present irrespective of time from most recent PCI. Interpretation: In patients with diabetes, stable coronary artery disease, and previous PCI, ticagrelor added to aspirin reduced cardiovascular death, myocardial infarction, and stroke, although with increased major bleeding. In that large, easily identified population, ticagrelor provided a favourable net clinical benefit (more than in patients without history of PCI). This effect shows that long-term therapy with ticagrelor in addition to aspirin should be considered in patients with diabetes and a history of PCI who have tolerated antiplatelet therapy, have high ischaemic risk, and low bleeding risk