Sums of Powers and the Bernoulli Numbers

This expository thesis examines the relationship between finite sums of powers and a sequence of numbers known as the Bernoulli numbers. It presents significant historical events tracing the discovery of formulas for finite sums of powers of integers, the discovery of a single formula by Jacob Bernoulli which gives the Bernoulli numbers, and important discoveries related to the Bernoulli numbers. A method of generating the sequence by means of a number theoretic recursive formula is given. Also given is an application of matrix theory to find a relation, first given by Johannes Faulhaber, between finite sums of odd powers and finite sums of even powers. An approach to finding a formula for sums of powers using integral calculus is also presented. The relation between the Bernoulli numbers and the coefficients of the Maclaurin expansion of f(z) = z /ez - 1, which was first given by Léonard Euler, is considered, as well as the trigonometric series expansions which are derived from the Maclaurin expansion of f(z), and the zeta function. Further areas of research relating to the topic are explored

Sums of Powers and the Bernoulli Numbers

This expository thesis examines the relationship between finite sums of powers and a sequence of numbers known as the Bernoulli numbers. It presents significant historical events tracing the discovery of formulas for finite sums of powers of integers, the discovery of a single formula by Jacob Bernoulli which gives the Bernoulli numbers, and important discoveries related to the Bernoulli numbers. A method of generating the sequence by means of a number theoretic recursive formula is given. Also given is an application of matrix theory to find a relation, first given by Johannes Faulhaber, between finite sums of odd powers and finite sums of even powers. An approach to finding a formula for sums of powers using integral calculus is also presented. The relation between the Bernoulli numbers and the coefficients of the Maclaurin expansion of f(z) = z /ez - 1, which was first given by Léonard Euler, is considered, as well as the trigonometric series expansions which are derived from the Maclaurin expansion of f(z), and the zeta function. Further areas of research relating to the topic are explored

Sums of Powers and the Bernoulli Numbers

This expository thesis examines the relationship between finite sums of powers and a sequence of numbers known as the Bernoulli numbers. It presents significant historical events tracing the discovery of formulas for finite sums of powers of integers, the discovery of a single formula by Jacob Bernoulli which gives the Bernoulli numbers, and important discoveries related to the Bernoulli numbers. A method of generating the sequence by means of a number theoretic recursive formula is given. Also given is an application of matrix theory to find a relation, first given by Johannes Faulhaber, between finite sums of odd powers and finite sums of even powers. An approach to finding a formula for sums of powers using integral calculus is also presented. The relation between the Bernoulli numbers and the coefficients of the Maclaurin expansion of f(z) = z /ez - 1, which was first given by Léonard Euler, is considered, as well as the trigonometric series expansions which are derived from the Maclaurin expansion of f(z), and the zeta function. Further areas of research relating to the topic are explored

Development of a Standard Reference Material for Metabolomics Research

The National Institute of Standards and Technology (NIST), in collaboration with the National Institutes of Health (NIH), has developed a Standard Reference Material (SRM) to support technology development in metabolomics research. SRM 1950 Metabolites in Human Plasma is intended to have metabolite concentrations that are representative of those found in adult human plasma. The plasma used in the preparation of SRM 1950 was collected from both male and female donors, and donor ethnicity targets were selected based upon the ethnic makeup of the U.S. population. Metabolomics research is diverse in terms of both instrumentation and scientific goals. This SRM was designed to apply broadly to the field, not towards specific applications. Therefore, concentrations of approximately 100 analytes, including amino acids, fatty acids, trace elements, vitamins, hormones, selenoproteins, clinical markers, and perfluorinated compounds (PFCs), were determined. Value assignment measurements were performed by NIST and the Centers for Disease Control and Prevention (CDC). SRM 1950 is the first reference material developed specifically for metabolomics research

A Survey of Empirical Results on Program Slicing

International audienceBACKGROUND:Patients with peripheral artery disease have an increased risk of cardiovascular morbidity and mortality. Antiplatelet agents are widely used to reduce these complications.METHODS:This was a multicentre, double-blind, randomised placebo-controlled trial for which patients were recruited at 602 hospitals, clinics, or community practices from 33 countries across six continents. Eligible patients had a history of peripheral artery disease of the lower extremities (previous peripheral bypass surgery or angioplasty, limb or foot amputation, intermittent claudication with objective evidence of peripheral artery disease), of the carotid arteries (previous carotid artery revascularisation or asymptomatic carotid artery stenosis of at least 50%), or coronary artery disease with an ankle-brachial index of less than 0·90. After a 30-day run-in period, patients were randomly assigned (1:1:1) to receive oral rivaroxaban (2·5 mg twice a day) plus aspirin (100 mg once a day), rivaroxaban twice a day (5 mg with aspirin placebo once a day), or to aspirin once a day (100 mg and rivaroxaban placebo twice a day). Randomisation was computer generated. Each treatment group was double dummy, and the patient, investigators, and central study staff were masked to treatment allocation. The primary outcome was cardiovascular death, myocardial infarction or stroke; the primary peripheral artery disease outcome was major adverse limb events including major amputation. This trial is registered with ClinicalTrials.gov, number NCT01776424, and is closed to new participants.FINDINGS:Between March 12, 2013, and May 10, 2016, we enrolled 7470 patients with peripheral artery disease from 558 centres. The combination of rivaroxaban plus aspirin compared with aspirin alone reduced the composite endpoint of cardiovascular death, myocardial infarction, or stroke (126 [5%] of 2492 vs 174 [7%] of 2504; hazard ratio [HR] 0·72, 95% CI 0·57-0·90, p=0·0047), and major adverse limb events including major amputation (32 [1%] vs 60 [2%]; HR 0·54 95% CI 0·35-0·82, p=0·0037). Rivaroxaban 5 mg twice a day compared with aspirin alone did not significantly reduce the composite endpoint (149 [6%] of 2474 vs 174 [7%] of 2504; HR 0·86, 95% CI 0·69-1·08, p=0·19), but reduced major adverse limb events including major amputation (40 [2%] vs 60 [2%]; HR 0·67, 95% CI 0·45-1·00, p=0·05). The median duration of treatment was 21 months. The use of the rivaroxaban plus aspirin combination increased major bleeding compared with the aspirin alone group (77 [3%] of 2492 vs 48 [2%] of 2504; HR 1·61, 95% CI 1·12-2·31, p=0·0089), which was mainly gastrointestinal. Similarly, major bleeding occurred in 79 (3%) of 2474 patients with rivaroxaban 5 mg, and in 48 (2%) of 2504 in the aspirin alone group (HR 1·68, 95% CI 1·17-2·40; p=0·0043).INTERPRETATION:Low-dose rivaroxaban taken twice a day plus aspirin once a day reduced major adverse cardiovascular and limb events when compared with aspirin alone. Although major bleeding was increased, fatal or critical organ bleeding was not. This combination therapy represents an important advance in the management of patients with peripheral artery disease. Rivaroxaban alone did not significantly reduce major adverse cardiovascular events compared with asprin alone, but reduced major adverse limb events and increased major bleeding

Outcomes in Newly Diagnosed Atrial Fibrillation and History of Acute Coronary Syndromes: Insights from GARFIELD-AF

BACKGROUND: Many patients with atrial fibrillation have concomitant coronary artery disease with or without acute coronary syndromes and are in need of additional antithrombotic therapy. There are few data on the long-term clinical outcome of atrial fibrillation patients with a history of acute coronary syndrome. This is a 2-year study of atrial fibrillation patients with or without a history of acute coronary syndromes

Development of the CMS detector for the CERN LHC Run 3

International audienceSince the initial data taking of the CERN LHC, the CMS experiment has undergone substantial upgrades and improvements. This paper discusses the CMS detector as it is configured for the third data-taking period of the CERN LHC, Run 3, which started in 2022. The entire silicon pixel tracking detector was replaced. A new powering system for the superconducting solenoid was installed. The electronics of the hadron calorimeter was upgraded. All the muon electronic systems were upgraded, and new muon detector stations were added, including a gas electron multiplier detector. The precision proton spectrometer was upgraded. The dedicated luminosity detectors and the beam loss monitor were refurbished. Substantial improvements to the trigger, data acquisition, software, and computing systems were also implemented, including a new hybrid CPU/GPU farm for the high-level trigger

Development of the CMS detector for the CERN LHC Run 3

International audienceSince the initial data taking of the CERN LHC, the CMS experiment has undergone substantial upgrades and improvements. This paper discusses the CMS detector as it is configured for the third data-taking period of the CERN LHC, Run 3, which started in 2022. The entire silicon pixel tracking detector was replaced. A new powering system for the superconducting solenoid was installed. The electronics of the hadron calorimeter was upgraded. All the muon electronic systems were upgraded, and new muon detector stations were added, including a gas electron multiplier detector. The precision proton spectrometer was upgraded. The dedicated luminosity detectors and the beam loss monitor were refurbished. Substantial improvements to the trigger, data acquisition, software, and computing systems were also implemented, including a new hybrid CPU/GPU farm for the high-level trigger

Development of the CMS detector for the CERN LHC Run 3

International audienceSince the initial data taking of the CERN LHC, the CMS experiment has undergone substantial upgrades and improvements. This paper discusses the CMS detector as it is configured for the third data-taking period of the CERN LHC, Run 3, which started in 2022. The entire silicon pixel tracking detector was replaced. A new powering system for the superconducting solenoid was installed. The electronics of the hadron calorimeter was upgraded. All the muon electronic systems were upgraded, and new muon detector stations were added, including a gas electron multiplier detector. The precision proton spectrometer was upgraded. The dedicated luminosity detectors and the beam loss monitor were refurbished. Substantial improvements to the trigger, data acquisition, software, and computing systems were also implemented, including a new hybrid CPU/GPU farm for the high-level trigger