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Brian Covello uses differential equations to provide an avenue for bridging the tautochrone and brachistochrone. The tautochrone problem, the attempt to identify this curve, was solved by Christiaan Huygens in 1659. He proved geometrically in his Horologium Oscillatorium, originally published in 1673, that the curve was a cycloid. On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after having departed from any point on the cycloid, are equal to each other...[1] Huygens also proved that the time of descent is equal to the time a body takes to fall vertically the same distance as the diameter of the circle which generates the cycloid, multiplied by π⁄2. In modern terms, this means that the time of descent is , where r is the radius of the circle which generates the cycloid and g is the gravity of Earth. This solution was later used to attack the problem of the brachistochrone curve. Jakob Bernoulli solved the problem using calculus in a paper (Acta Eruditorum, 1690) that saw the first published use of the term integral.[2] Schematic of a cycloidal pendulum. The tautochrone problem was studied more closely when it was realized that a pendulum, which follows a circular path, was not isochronous and thus his pendulum clock would keep different time depending on how far the pendulum swung. After determining the correct path, Christiaan Huygens attempted to create pendulum clocks that used a string to suspend the bob and curb cheeks near the top of the string to change the path to the tautochrone curve. These attempts proved to not be useful for a number of reasons. First, the bending of the string causes friction, changing the timing. Second, there were much more significant sources of timing errors that overwhelmed any theoretical improvements that traveling on the tautochrone curve helps. Finally, the "circular error" of a pendulum decreases as length of the swing decreases, so better clock escapements could greatly reduce this source of inaccuracy. Later, the mathematicians Joseph Louis Lagrange and Leonhard Euler provided an analytical solution to the problem. A brachistochrone curve (Gr. βράχιστος, brachistos - the shortest, χρόνος, chronos - time) or curve of fastest descent, is the path that will carry a point-like body from one place to another in the least amount of time. The body is released at rest from the starting point and is constrained to move without friction along the curve to the end point, while under the action of constant gravity. The brachistochrone curve is the same as the tautochrone curve for a given starting point. Given two points A and B, with A not lower than B, only one upside down cycloid passes through both points, has a vertical tangent line at A, and has no maximum points between A and B: the brachistochrone curve.

Brian Covello: Review on Cycloidal Pathways Using Differential Equations

Brian Covello

Brian Covello's diabetes research proposal. Type 2 diabetes mellitus consists of an array of dysfunctions characterized by hyperglycemia and resulting from the combination of resistance to insulin action, inadequate insulin secretion, and excessive or inappropriate glucagon secretion. Essential update: FDA approves subcutaneous albiglutide for management of DM2 The FDA has approved once-weekly injectable albiglutide (Tanzeum), a glucagonlike peptide 1 (GLP-1) receptor agonist, along with diet and exercise for the treatment of type 2 diabetes.[1, 2] This agent may be used either as monotherapy or in combination with metformin, glimepiride, pioglitazone, or insulin. Albiglutide should not be used for the following[1, 2] : Patients with type 1 diabetes Patients with diabetic ketoacidosis First-line therapy in patients who can’t be managed with diet and exercise Patients who have a personal or family history of medullary thyroid carcinoma (MTC) Patients who have multiple endocrine neoplasia syndrome type 2 The most common adverse reactions associated with albiglutide were nausea/diarrhea and injection-site reactions. There will be a boxed warning on albiglutide’s labeling about thyroid C-cell tumors being observed in rodent studies with this class of drugs; it is currently unknown whether albiglutide causes these tumors in humans, including MTC.[1, 2] Moreover, the FDA is also requiring a number of postmarketing studies, including a pediatric trial; an MTC case registry (≥15 y); and a cardiovascular (CV)-outcomes trial in patients with a baseline high risk of CV disease. Signs and symptoms Many patients with type 2 diabetes are asymptomatic. Clinical manifestations include the following: Classic symptoms: Polyuria, polydipsia, polyphagia, and weight loss Blurred vision Lower-extremity paresthesias Yeast infections (eg, balanitis in men) See Presentation for more detail. Diagnosis Diagnostic criteria by the American Diabetes Association (ADA) include the following[3] : A fasting plasma glucose (FPG) level of 126 mg/dL (7.0 mmol/L) or higher, or A 2-hour plasma glucose level of 200 mg/dL (11.1 mmol/L) or higher during a 75-g oral glucose tolerance test (OGTT), or A random plasma glucose of 200 mg/dL (11.1 mmol/L) or higher in a patient with classic symptoms of hyperglycemia or hyperglycemic crisis Whether a hemoglobin A1c (HbA1c) level of 6.5% or higher should be a primary diagnostic criterion or an optional criterion remains a point of controversy. Indications for diabetes screening in asymptomatic adults includes the following[4, 5] : Sustained blood pressure >135/80 mm Hg Overweight and 1 or more other risk factors for diabetes (eg, first-degree relative with diabetes, BP >140/90 mm Hg, and HDL < 35 mg/dL and/or triglyceride level >250 mg/dL) ADA recommends screening at age 45 years in the absence of the above criteria See Workup for more detail.

Brian Covello: Diabetes Research Proposal

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Brian Covello's work with the isochronous pendulum. Taken from www.wikipedia.org A pendulum is a weight suspended from a pivot so that it can swing freely.[1] When a pendulum is displaced sideways from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum, and also to a slight degree on the amplitude, the width of the pendulum's swing. From its discovery in around 1602 by Galileo Galilei, the regular motion of pendulums was used for timekeeping, and was the world's most accurate timekeeping technology until the 1930s.[2] Pendulums are used to regulate pendulum clocks, and are used in scientific instruments such as accelerometers and seismometers. Historically they were used as gravimeters to measure the acceleration of gravity in geophysical surveys, and even as a standard of length. The word 'pendulum' is new Latin, from the Latin pendulus, meaning 'hanging'.[3] The simple gravity pendulum[4] is an idealized mathematical model of a pendulum.[5][6][7] This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. When given an initial push, it will swing back and forth at a constant amplitude. Real pendulums are subject to friction and air drag, so the amplitude of their swings declines.

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