Formally we mean that is an ideal if it satisfies the following conditions:

Formally, if we write F < sub > Δ </ sub >( x ) to mean the f-polynomial of Δ, then the h-polynomial of Δ is

Formally, the Mahalanobis distance of a multivariate vector from a group of values with mean and covariance matrix is defined as:

Formally, a hypergraph is a pair where is a set of elements called nodes or vertices, and is a set of non-empty subsets of called hyperedges or edges.

Formally, a non-empty partially ordered set ( D, ≤) is called a Scott domain if the following hold:

Formally, a binary operation on a set S is called associative if it satisfies the associative law:

Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).

Formally speaking, a collation method typically defines a total order on a set of possible identifiers, called sort keys, which consequently produces a total preorder on the set of items of information ( items with the same identifier are not placed in any defined order ).

Formally, one has an implication, not an equivalence, so the solution set may get larger.

Formally, a set S is called finite if there exists a bijection

Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies

Formally, according to the Constitution, citizens of Turkmenistan have the right to set up political parties and other public associations, acting within the framework of the Constitution and laws, and public associations and groups of citizens have the right to nominate their candidates in accordance with the election law.

Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X.

Formally, Gaussian integers are the set

Formally, the discrete cosine transform is a linear, invertible function ( where denotes the set of real numbers ), or equivalently an invertible N × N square matrix.

Formally, we are given a set of hypotheses and a set of manifestations ; they are related by the domain knowledge, represented by a function that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations.

Formally, a function ƒ is real analytic on an open set D in the real line if for any x < sub > 0 </ sub > in D one can write

Formally, start with a set Ω and consider the sigma algebra Σ on Ω consisting of all subsets of Ω.

Formally, the movement is a rondo that acts as the theme for a set of eight variations, capped off by a dramatic coda.

Formally, the discrete sine transform is a linear, invertible function F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup > ( where R denotes the set of real numbers ), or equivalently an N × N square matrix.

Formally, starting with a linearly independent set of vectors

Formally, the discrete Hartley transform is a linear, invertible function H: R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup > ( where R denotes the set of real numbers ).

Formally, a constraint satisfaction problem is defined as a triple, where is a set of variables, is a domain of values, and is a set of constraints.

Formally, the problem of supervised pattern recognition can be stated as follows: Given an unknown function ( the ground truth ) that maps input instances to output labels, along with training data assumed to represent accurate examples of the mapping, produce a function that approximates as closely as possible the correct mapping.

Formally, the outcomes Y < sub > i </ sub > are described as being Bernoulli-distributed data, where each outcome is determined by an unobserved probability p < sub > i </ sub > that is specific to the outcome at hand, but related to the explanatory variables.

Formally, citizen science has been defined as " the systematic collection and analysis of data ; development of technology ; testing of natural phenomena ; and the dissemination of these activities by researchers on a primarily avocational basis ".

* Trimean – the weighted arithmetic mean of the median and two quartiles.

In probability theory, the expected value ( or expectation, or mathematical expectation, or mean, or the first moment ) of a random variable is the weighted average of all possible values that this random variable can take on.

Non-investable indices are indicative in nature, and aim to represent the performance of some database of hedge funds using some measure such as mean, median or weighted mean from a hedge fund database.

If a set of weights, ..., is associated to the dataset, ...,, the weighted harmonic mean is defined by

It is also equivalent to any weighted harmonic mean where all weights are equal.

( If neither is the case, then a weighted harmonic mean or weighted arithmetic mean is needed.

The weighted harmonic mean is the correct approach to determine the average specific gravity of a mixture when the composition by weight is known.

For a probability distribution, the mean is equal to the sum or integral over every possible value weighted by the probability of that value.

The average molecular mass is the abundance weighted mean ( average ) of the molecular masses in a sample.

It can produce a weighted mean that has less variability than the arithmetic mean of a simple random sample of the population.

The weighted mean is similar to an arithmetic mean ( the most common type of average ), where instead of each of the data points contributing equally to the final average, some data points contribute more than others.

The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

If all the weights are equal, then the weighted mean is the same as the arithmetic mean.

The term weighted average usually refers to a weighted arithmetic mean, but weighted versions of other means can also be calculated, such as the weighted geometric mean and the weighted harmonic mean.

* Input: A non-empty connected weighted graph with vertices V and edges E ( the weights can be negative ).