Tensor product of modulesIn mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group.
Module (mathematics)In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication.
Free moduleIn mathematics, a free module is a module that has a basis, that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S. A free abelian group is precisely a free module over the ring Z of integers.
Projective moduleIn mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains.
DNA microarrayA DNA microarray (also commonly known as DNA chip or biochip) is a collection of microscopic DNA spots attached to a solid surface. Scientists use DNA microarrays to measure the expression levels of large numbers of genes simultaneously or to genotype multiple regions of a genome. Each DNA spot contains picomoles (10−12 moles) of a specific DNA sequence, known as probes (or reporters or oligos). These can be a short section of a gene or other DNA element that are used to hybridize a cDNA or cRNA (also called anti-sense RNA) sample (called target) under high-stringency conditions.
Gene expression profilingIn the field of molecular biology, gene expression profiling is the measurement of the activity (the expression) of thousands of genes at once, to create a global picture of cellular function. These profiles can, for example, distinguish between cells that are actively dividing, or show how the cells react to a particular treatment. Many experiments of this sort measure an entire genome simultaneously, that is, every gene present in a particular cell. Several transcriptomics technologies can be used to generate the necessary data to analyse.
Microarray analysis techniquesMicroarray analysis techniques are used in interpreting the data generated from experiments on DNA (Gene chip analysis), RNA, and protein microarrays, which allow researchers to investigate the expression state of a large number of genes - in many cases, an organism's entire genome - in a single experiment. Such experiments can generate very large amounts of data, allowing researchers to assess the overall state of a cell or organism. Data in such large quantities is difficult - if not impossible - to analyze without the help of computer programs.
Bayesian statisticsBayesian statistics (ˈbeɪziən or ˈbeɪʒən ) is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a degree of belief in an event. The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal beliefs about the event. This differs from a number of other interpretations of probability, such as the frequentist interpretation that views probability as the limit of the relative frequency of an event after many trials.
Ovarian cancerOvarian cancer is a cancerous tumor of an ovary. It may originate from the ovary itself or more commonly from communicating nearby structures such as fallopian tubes or the inner lining of the abdomen. The ovary is made up of three different cell types including epithelial cells, germ cells, and stromal cells. When these cells become abnormal, they have the ability to divide and form tumors. These cells can also invade or spread to other parts of the body. When this process begins, there may be no or only vague symptoms.
Summary statisticsIn descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount of information as simply as possible. Statisticians commonly try to describe the observations in a measure of location, or central tendency, such as the arithmetic mean a measure of statistical dispersion like the standard mean absolute deviation a measure of the shape of the distribution like skewness or kurtosis if more than one variable is measured, a measure of statistical dependence such as a correlation coefficient A common collection of order statistics used as summary statistics are the five-number summary, sometimes extended to a seven-number summary, and the associated box plot.
SimulationA simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the simulation represents the evolution of the model over time. Often, computers are used to execute the simulation. Simulation is used in many contexts, such as simulation of technology for performance tuning or optimizing, safety engineering, testing, training, education, and video games.
Sheaf of modulesIn mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) → F(V) are compatible with the restriction maps O(U) → O(V): the restriction of fs is the restriction of f times that of s for any f in O(U) and s in F(U). The standard case is when X is a scheme and O its structure sheaf. If O is the constant sheaf , then a sheaf of O-modules is the same as a sheaf of abelian groups (i.
Descriptive statisticsA descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and analysing those statistics. Descriptive statistics is distinguished from inferential statistics (or inductive statistics) by its aim to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent.
Bayesian hierarchical modelingBayesian hierarchical modelling is a statistical model written in multiple levels (hierarchical form) that estimates the parameters of the posterior distribution using the Bayesian method. The sub-models combine to form the hierarchical model, and Bayes' theorem is used to integrate them with the observed data and account for all the uncertainty that is present. The result of this integration is the posterior distribution, also known as the updated probability estimate, as additional evidence on the prior distribution is acquired.
Finitely generated moduleIn mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide.
Prior probabilityA prior probability distribution of an uncertain quantity, often simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable.
Five-number summaryThe five-number summary is a set of descriptive statistics that provides information about a dataset. It consists of the five most important sample percentiles: the sample minimum (smallest observation) the lower quartile or first quartile the median (the middle value) the upper quartile or third quartile the sample maximum (largest observation) In addition to the median of a single set of data there are two related statistics called the upper and lower quartiles.
Jeffreys priorIn Bayesian probability, the Jeffreys prior, named after Sir Harold Jeffreys, is a non-informative prior distribution for a parameter space; its density function is proportional to the square root of the determinant of the Fisher information matrix: It has the key feature that it is invariant under a change of coordinates for the parameter vector . That is, the relative probability assigned to a volume of a probability space using a Jeffreys prior will be the same regardless of the parameterization used to define the Jeffreys prior.
Seven-number summaryIn descriptive statistics, the seven-number summary is a collection of seven summary statistics, and is an extension of the five-number summary. There are three similar, common forms. As with the five-number summary, it can be represented by a modified box plot, adding hatch-marks on the "whiskers" for two of the additional numbers. The following percentiles are (approximately) evenly spaced under a normally distributed variable: the 2nd percentile (better: 2.15%) the 9th percentile (better: 8.
Divisible groupIn mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups. An abelian group is divisible if, for every positive integer and every , there exists such that .
