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MATHICSE Technical Report : Frequency-domain non-intrusive greedy Model Order Reduction based on minimal rational approximation

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Liouville number

In number theory, a Liouville number is a real number with the property that, for every positive integer , there exists a pair of integers with such that Liouville numbers are "almost rational", and can thus be approximated "quite closely" by sequences of rational numbers. Precisely, these are transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number can be. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time.

Logistic regression

In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression analysis, logistic regression (or logit regression) is estimating the parameters of a logistic model (the coefficients in the linear combination).

Pearson's chi-squared test

Pearson's chi-squared test () is a statistical test applied to sets of categorical data to evaluate how likely it is that any observed difference between the sets arose by chance. It is the most widely used of many chi-squared tests (e.g., Yates, likelihood ratio, portmanteau test in time series, etc.) – statistical procedures whose results are evaluated by reference to the chi-squared distribution. Its properties were first investigated by Karl Pearson in 1900.

Eigenvalues and eigenvectors

In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.

Vector-valued differential form

In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.) Let M be a smooth manifold and E → M be a smooth vector bundle over M.

Integral transform

In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the inverse transform. An integral transform is any transform of the following form: The input of this transform is a function , and the output is another function .

Transcendental number theory

Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways. Transcendental number The fundamental theorem of algebra tells us that if we have a non-constant polynomial with rational coefficients (or equivalently, by clearing denominators, with integer coefficients) then that polynomial will have a root in the complex numbers.

Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space . A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space (for example could be a topological space, a manifold, or an algebraic variety): to every point of the space we associate (or "attach") a vector space in such a way that these vector spaces fit together to form another space of the same kind as (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over .