PolynomialIn mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1. Polynomials appear in many areas of mathematics and science.
Dirichlet's approximation theoremIn number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers and , with , there exist integers and such that and Here represents the integer part of . This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality is satisfied by infinitely many integers p and q.
Separable polynomialIn mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots is equal to the degree of the polynomial. This concept is closely related to square-free polynomial. If K is a perfect field then the two concepts coincide. In general, P(X) is separable if and only if it is square-free over any field that contains K, which holds if and only if P(X) is coprime to its formal derivative D P(X).
Ensemble Kalman filterThe ensemble Kalman filter (EnKF) is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models. The EnKF originated as a version of the Kalman filter for large problems (essentially, the covariance matrix is replaced by the sample covariance), and it is now an important data assimilation component of ensemble forecasting.
Bravais latticeIn geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by where the ni are any integers, and ai are primitive translation vectors, or primitive vectors, which lie in different directions (not necessarily mutually perpendicular) and span the lattice. The choice of primitive vectors for a given Bravais lattice is not unique.
Self-organizing mapA self-organizing map (SOM) or self-organizing feature map (SOFM) is an unsupervised machine learning technique used to produce a low-dimensional (typically two-dimensional) representation of a higher dimensional data set while preserving the topological structure of the data. For example, a data set with variables measured in observations could be represented as clusters of observations with similar values for the variables.
Separable spaceIn mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense.
Lattice (group)In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
Deep learningDeep learning is part of a broader family of machine learning methods, which is based on artificial neural networks with representation learning. The adjective "deep" in deep learning refers to the use of multiple layers in the network. Methods used can be either supervised, semi-supervised or unsupervised.
Reciprocal latticeIn physics, the reciprocal lattice represents the Fourier transform of another lattice. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system (usually a Bravais lattice). The reciprocal lattice exists in the mathematical space of spatial frequencies, known as reciprocal space or k space, where refers to the wavevector. In quantum physics, reciprocal space is closely related to momentum space according to the proportionality , where is the momentum vector and is the reduced Planck constant.
WaveletA wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly one tenth of a second.
Algebraic closureIn mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemma or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.
