DerivativeIn mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
Β-Galactosidaseβ-Galactosidase (EC 3.2.1.23, lactase, beta-gal or β-gal; systematic name β-D-galactoside galactohydrolase), is a glycoside hydrolase enzyme that catalyzes hydrolysis of terminal non-reducing β-D-galactose residues in β-D-galactosides. β-Galactosides include carbohydrates containing galactose where the glycosidic bond lies above the galactose molecule. Substrates of different β-galactosidases include ganglioside GM1, lactosylceramides, lactose, and various glycoproteins.
Generalizations of the derivativeIn mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. The Fréchet derivative defines the derivative for general normed vector spaces . Briefly, a function , an open subset of , is called Fréchet differentiable at if there exists a bounded linear operator such that Functions are defined as being differentiable in some open neighbourhood of , rather than at individual points, as not doing so tends to lead to many pathological counterexamples.
Second derivativeIn calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation: where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change.
Derivative testIn calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is proved mathematically by Fermat's theorem of stationary points. The first-derivative test examines a function's monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain.
Logarithmic derivativeIn mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely scaled by the current value of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f.
DisaccharideA disaccharide (also called a double sugar or biose) is the sugar formed when two monosaccharides are joined by glycosidic linkage. Like monosaccharides, disaccharides are simple sugars soluble in water. Three common examples are sucrose, lactose, and maltose. Disaccharides are one of the four chemical groupings of carbohydrates (monosaccharides, disaccharides, oligosaccharides, and polysaccharides). The most common types of disaccharides—sucrose, lactose, and maltose—have 12 carbon atoms, with the general formula C12H22O11.
Partial derivativeIn mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function with respect to the variable is variously denoted by It can be thought of as the rate of change of the function in the -direction.
Covariant derivativeIn mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space.
CarbohydrateA carbohydrate (ˌkɑːrboʊˈhaɪdreɪt) is a biomolecule consisting of carbon (C), hydrogen (H) and oxygen (O) atoms, usually with a hydrogen–oxygen atom ratio of 2:1 (as in water) and thus with the empirical formula (where m may or may not be different from n), which does not mean the H has covalent bonds with O (for example with , H has a covalent bond with C but not with O). However, not all carbohydrates conform to this precise stoichiometric definition (e.g.
Reducing sugarA reducing sugar is any sugar that is capable of acting as a reducing agent. In an alkaline solution, a reducing sugar forms some aldehyde or ketone, which allows it to act as a reducing agent, for example in Benedict's reagent. In such a reaction, the sugar becomes a carboxylic acid. All monosaccharides are reducing sugars, along with some disaccharides, some oligosaccharides, and some polysaccharides. The monosaccharides can be divided into two groups: the aldoses, which have an aldehyde group, and the ketoses, which have a ketone group.
Chemical synthesisIn chemistry, chemical synthesis (chemical combination) is the artificial execution of chemical reactions to obtain one or several products. This occurs by physical and chemical manipulations usually involving one or more reactions. In modern laboratory uses, the process is reproducible and reliable. A chemical synthesis involves one or more compounds (known as reagents or reactants) that will experience a transformation when subjected to certain conditions. Various reaction types can be applied to formulate a desired product.
Total synthesisTotal synthesis is the complete chemical synthesis of a complex molecule, often a natural product, from simple, commercially-available precursors. It usually refers to a process not involving the aid of biological processes, which distinguishes it from semisynthesis. Syntheses may sometimes conclude at a precursor with further known synthetic pathways to a target molecule, in which case it is known as a formal synthesis. Total synthesis target molecules can be natural products, medicinally-important active ingredients, known intermediates, or molecules of theoretical interest.
Glycoside hydrolaseGlycoside hydrolases (also called glycosidases or glycosyl hydrolases) catalyze the hydrolysis of glycosidic bonds in complex sugars. They are extremely common enzymes with roles in nature including degradation of biomass such as cellulose (cellulase), hemicellulose, and starch (amylase), in anti-bacterial defense strategies (e.g., lysozyme), in pathogenesis mechanisms (e.g., viral neuraminidases) and in normal cellular function (e.g., trimming mannosidases involved in N-linked glycoprotein biosynthesis).
GlycosaminoglycanGlycosaminoglycans (GAGs) or mucopolysaccharides are long, linear polysaccharides consisting of repeating disaccharide units (i.e. two-sugar units). The repeating two-sugar unit consists of a uronic sugar and an amino sugar, except in the case of the sulfated glycosaminoglycan keratan, where, in place of the uronic sugar there is a galactose unit. GAGs are found in vertebrates, invertebrates and bacteria. Because GAGs are highly polar molecules and attract water; the body uses them as lubricants or shock absorbers.
K-theoryIn mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.
Keratan sulfateKeratan sulfate (KS), also called keratosulfate, is any of several sulfated glycosaminoglycans (structural carbohydrates) that have been found especially in the cornea, cartilage, and bone. It is also synthesized in the central nervous system where it participates both in development and in the glial scar formation following an injury. Keratan sulfates are large, highly hydrated molecules which in joints can act as a cushion to absorb mechanical shock.
States' rightsIn American political discourse, states' rights are political powers held for the state governments rather than the federal government according to the United States Constitution, reflecting especially the enumerated powers of Congress and the Tenth Amendment. The enumerated powers that are listed in the Constitution include exclusive federal powers, as well as concurrent powers that are shared with the states, and all of those powers are contrasted with the reserved powers—also called states' rights—that only the states possess.
Algebraic K-theoryAlgebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.
BeanA bean is the seed of several plants in the family Fabaceae, which are used as vegetables for human or animal food. They can be cooked in many different ways, including boiling, frying, and baking, and are used in many traditional dishes throughout the world. The word "bean" and its Germanic cognates (e.g. German Bohne) have existed in common use in West Germanic languages since before the 12th century, referring to broad beans, chickpeas, and other pod-borne seeds. This was long before the New World genus Phaseolus was known in Europe.
