Polynomial interpolationIn numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of n + 1 data points , with no two the same, a polynomial function is said to interpolate the data if for each . There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials.
Legendre polynomialsIn mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions.
Lagrange polynomialIn numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs with the are called nodes and the are called values. The Lagrange polynomial has degree and assumes each value at the corresponding node, Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler.
Bile acidBile acids are steroid acids found predominantly in the bile of mammals and other vertebrates. Diverse bile acids are synthesized in the liver. Bile acids are conjugated with taurine or glycine residues to give anions called bile salts. Primary bile acids are those synthesized by the liver. Secondary bile acids result from bacterial actions in the colon. In humans, taurocholic acid and glycocholic acid (derivatives of cholic acid) and taurochenodeoxycholic acid and glycochenodeoxycholic acid (derivatives of chenodeoxycholic acid) are the major bile salts.
Orthogonal polynomialsIn mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases.
BileBile (from Latin bilis), or gall, is a yellow-green fluid produced by the liver of most vertebrates that aids the digestion of lipids in the small intestine. In humans, bile is primarily composed of water, produced continuously by the liver, and stored and concentrated in the gallbladder. After a human eats, this stored bile is discharged into the first section of their small intestine. In the human liver, bile is composed of 97–98% water, 0.7% bile salts, 0.2% bilirubin, 0.
Proof by contradictionIn logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.
Bile ductA bile duct is any of a number of long tube-like structures that carry bile, and is present in most vertebrates. Bile is required for the digestion of food and is secreted by the liver into passages that carry bile toward the hepatic duct. It joins the cystic duct (carrying bile to and from the gallbladder) to form the common bile duct which then opens into the intestine. The top half of the common bile duct is associated with the liver, while the bottom half of the common bile duct is associated with the pancreas, through which it passes on its way to the intestine.
Common bile ductThe bile duct (formerly known as the common bile duct) is a part of the biliary tract. It is formed by the union of the common hepatic duct and cystic duct. It ends by uniting with the pancreatic duct to form the hepatopancreatic ampulla. It possesses its own sphincter to enable regulation of bile flow. The bile duct is some 6–8 cm long, and normally up to 8 mm in diameter. Its proximal supraduodenal part is situated within the free edge of the lesser omentum.
CholangiocarcinomaCholangiocarcinoma, also known as bile duct cancer, is a type of cancer that forms in the bile ducts. Symptoms of cholangiocarcinoma may include abdominal pain, yellowish skin, weight loss, generalized itching, and fever. Light colored stool or dark urine may also occur. Other biliary tract cancers include gallbladder cancer and cancer of the ampulla of Vater. Risk factors for cholangiocarcinoma include primary sclerosing cholangitis (an inflammatory disease of the bile ducts), ulcerative colitis, cirrhosis, hepatitis C, hepatitis B, infection with certain liver flukes, and some congenital liver malformations.
Associated Legendre polynomialsIn mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation or equivalently where the indices l and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on only if l and m are integers with 0 ≤ m ≤ l, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial.
Power seriesIn mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, c (the center of the series) is equal to zero, for instance when considering a Maclaurin series.
Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Taylor seriesIn mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.
Proof by exhaustionProof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion typically contains two stages: A proof that the set of cases is exhaustive; i.e.
Proof (truth)A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition.
Obeticholic acidObeticholic acid (OCA), sold under the brand name Ocaliva, is a semi-synthetic bile acid analogue which has the chemical structure 6α-ethyl-chenodeoxycholic acid. It is used as a medication used to treat primary biliary cholangitis. Intercept Pharmaceuticals Inc. hold the worldwide rights to develop OCA outside Japan and China, where it is licensed to Dainippon Sumitomo Pharma. The natural bile acid chenodeoxycholic acid was identified in 1999 as the most active physiological ligand for the farnesoid X receptor (FXR), which is involved in many physiological and pathological processes.
Madhava seriesIn mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century Kerala by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. Using modern notation, these series are: All three series were later independently discovered in 17th century Europe.
Series (mathematics)In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.
X-ray tubeAn X-ray tube is a vacuum tube that converts electrical input power into X-rays. The availability of this controllable source of X-rays created the field of radiography, the imaging of partly opaque objects with penetrating radiation. In contrast to other sources of ionizing radiation, X-rays are only produced as long as the X-ray tube is energized. X-ray tubes are also used in CT scanners, airport luggage scanners, X-ray crystallography, material and structure analysis, and for industrial inspection.
