Symmetric groupIn abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are ( factorial) such permutation operations, the order (number of elements) of the symmetric group is .
CaffeineCaffeine is a central nervous system (CNS) stimulant of the methylxanthine class. It is mainly used recreationally, as a eugeroic (wakefulness promoter) or as a mild cognitive enhancer to increase alertness and attentional performance. Caffeine acts by blocking binding of adenosine to the adenosine A1 receptor, which enhances release of the neurotransmitter acetylcholine. Caffeine has a three-dimensional structure similar to that of adenosine, which allows it to bind and block its receptors.
Representation theory of the symmetric groupIn mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids. The symmetric group Sn has order n!. Its conjugacy classes are labeled by partitions of n.
MorphineMorphine is a strong opiate that is found naturally in opium, a dark brown resin produced by drying the latex of opium poppies (Papaver somniferum). It is mainly used as a pain medication, and is also commonly used recreationally, or to make other illicit opioids. There are numerous methods used to administer morphine: oral; sublingual; via inhalation; injection into a muscle, injection under the skin, or injection into the spinal cord area; transdermal; or via rectal suppository.
AmphetamineAmphetamine (contracted from alpha-methylphenethylamine) is a central nervous system (CNS) stimulant that is used in the treatment of attention deficit hyperactivity disorder (ADHD), narcolepsy, and obesity. Amphetamine was discovered in 1887 and exists as two enantiomers: levoamphetamine and dextroamphetamine. Amphetamine properly refers to a specific chemical, the racemic free base, which is equal parts of the two enantiomers in their pure amine forms.
Tensor productIn mathematics, the tensor product of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map that maps a pair to an element of denoted An element of the form is called the tensor product of v and w. An element of is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span in the sense that every element of is a sum of elementary tensors.
Bilinear formIn mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The dot product on is an example of a bilinear form.
Chemical reactionA chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking of chemical bonds between atoms, with no change to the nuclei (no change to the elements present), and can often be described by a chemical equation. Nuclear chemistry is a sub-discipline of chemistry that involves the chemical reactions of unstable and radioactive elements where both electronic and nuclear changes can occur.
PeptidePeptides are short chains of amino acids linked by peptide bonds. A polypeptide is a longer, continuous, unbranched peptide chain. Polypeptides which have a molecular mass of 10,000 Da or more are called proteins. Chains of fewer than twenty amino acids are called oligopeptides, and include dipeptides, tripeptides, and tetrapeptides. Peptides fall under the broad chemical classes of biological polymers and oligomers, alongside nucleic acids, oligosaccharides, polysaccharides, and others.
Rate equationIn chemistry, the rate law or rate equation for a chemical reaction is a mathematical equation that links the rate of forward reaction with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reaction orders). For many reactions, the initial rate is given by a power law such as where [\mathrm{A}] and [\mathrm{B}] express the concentration of the species \mathrm{A} and \mathrm{B}, usually in moles per liter (molarity, M).
