Kernel (algebra)In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix. The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element.
Professional degreeA professional degree, formerly known in the US as a first professional degree, is a degree that prepares someone to work in a particular profession, practice, or industry sector often meeting the academic requirements for licensure or accreditation. Professional degrees may be either graduate or undergraduate entry, depending on the profession concerned and the country, and may be classified as bachelor's, master's, or doctoral degrees.
Kernel (set theory)In set theory, the kernel of a function (or equivalence kernel) may be taken to be either the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function can tell", or the corresponding partition of the domain. An unrelated notion is that of the kernel of a non-empty family of sets which by definition is the intersection of all its elements: This definition is used in the theory of filters to classify them as being free or principal.
Elementary algebraElementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entails use of algebraic notation and an understanding of the general rules of the operations introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.
Circulant matrixIn linear algebra, a circulant matrix is a square matrix in which all row vectors are composed of the same elements and each row vector is rotated one element to the right relative to the preceding row vector. It is a particular kind of Toeplitz matrix. In numerical analysis, circulant matrices are important because they are diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform.
Weierstrass transformIn mathematics, the Weierstrass transform of a function f : R → R, named after Karl Weierstrass, is a "smoothed" version of f(x) obtained by averaging the values of f, weighted with a Gaussian centered at x. Specifically, it is the function F defined by the convolution of f with the Gaussian function The factor 1/√(4π) is chosen so that the Gaussian will have a total integral of 1, with the consequence that constant functions are not changed by the Weierstrass transform. Instead of F(x) one also writes Wf. 