As an algebra, a unital commutative Banach algebra is semisimple ( i. e., its Jacobson radical is zero ) if and only if its Gelfand representation has trivial kernel.

: A faithful representation is one in which the homomorphism G → GL ( V ) is injective ; in other words, one whose kernel is the trivial subgroup

A CAD system can be seen as built up from the interaction of a graphical user interface ( GUI ) with NURBS geometry and / or boundary representation ( B-rep ) data via a geometric modeling kernel.

Now, the representation is faithful if is injective, that is, if the kernel of is trivial.

a scale space representation obtained by smoothing the original image with a Gaussian kernel.

However, one can lift a projective representation of G to a linear representation of a different group C, which will be a central extension of G. To understand this, note that GL ( V ) → PGL ( V ) is a central extension of PGL, meaning that the kernel is central ( in fact, is exactly the center of GL ).

As a consequence, the kernel of the Gelfand representation A → C < sub > 0 </ sub >( Φ < sub > A </ sub >) may be identified with the Jacobson radical of A.

However, this is no longer a strictly time – frequency representation – the kernel is not constant over the entire signal.

The kernel is designed to provide support to persistently store highly granular nodes of knowledge representation like terms, predicates and very complex propositional systems like arguments, rules, axiomatic systems, loosely held paragraphs, and more complex structured and consistent compositions.

These techniques construct a low-dimensional data representation using a cost function that retains local properties of the data, and can be viewed as defining a graph-based kernel for Kernel PCA.

This function is called the reproducing kernel for the Hilbert space H and it is determined entirely by H because the Riesz representation theorem guarantees, for every x in X, that the element K < sub > x </ sub > satisfying (*) is unique.

When G is finite and F has characteristic zero, the kernel of the character χ < sub > ρ </ sub > is the normal subgroup:, which is precisely the kernel of the representation ρ.

Conversely, if G contains a proper non-trivial normal subgroup N, then the composition of the natural surjective group homomorphism G → G / N with the regular representation of G / N produces a representation π of G which has kernel N. Taking χ to be the character of some non-trivial subrepresentation of π, we have a character satisfying the hypothesis in the direct statement above.

For a finite, p-constrained group, an irreducible module over a field of characteristic p lies in the principal block if and only if the p ′- core of the group is contained in the kernel of the representation.

As a built-in pre-requisite to feature detection, the input image is usually smoothed by a Gaussian kernel in a scale-space representation and one or several feature images are computed, often expressed in terms of local derivative operations.

It is a formal theory for handling image structures at different scales, by representing an image as a one-parameter family of smoothed images, the scale-space representation, parametrized by the size of the smoothing kernel used for suppressing fine-scale structures.

For a given image, its linear ( Gaussian ) scale-space representation is a family of derived signals defined by the convolution of with the Gaussian kernel

Interestingly, the uniqueness of the Gaussian derivative operators as local operations derived from a scale-space representation can be obtained by similar axiomatic derivations as are used for deriving the uniqueness of the Gaussian kernel for scale-space smoothing.

Every Fredholm kernel has a representation in the form

Although ρ < sub > n </ sub > has a non-trivial kernel, the nonzero elements of that kernel have degree at least ( they are multiples of X < sub > 1 </ sub > X < sub > 2 </ sub >… X < sub > n + 1 </ sub >).

If G is not itself a Lie group, there must be a kernel to ρ.

Further one can form an inverse system, for the kernel of ρ smaller and smaller, of finite-dimensional unitary representations, which identifies G as an inverse limit of compact Lie groups.

A consequence of this is that if χ is a non-trivial irreducible character of G such that χ ( g ) = χ ( 1 ) for some g ≠ 1 then G contains a proper non-trivial normal subgroup ( the normal subgroup is the kernel of ρ ).

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism is isomorphic to where ker ( φ ) denotes the kernel of φ.

Development on the Hurd began in 1990 after an abandoned kernel attempt in 1986, based on the research TRIX operating system developed by Professor Steve Ward and his group at MIT's Laboratory for Computer Science ( LCS ).

To understand the concept, it is useful to study several examples first, of which there are many: all free objects, direct product and direct sum, free group, free lattice, Grothendieck group, product topology, Stone – Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.

There is a convenient relationship between the kernel and cokernel and the Abelian group structure on the hom-sets.

Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.

The underpinnings for the contemporary moral panic were found in a rise of five factors in the years leading up to the 1980s: The establishment of Fundamentalist Christianity and political organization of the Moral Majority ; the rise of the Anti-cult movement which spread ideas of abusive cults kidnapping and brainwashing children and teens ; the appearance of the Church of Satan and other explicitly Satanist groups that added a kernel of truth to the existence of Satanic cults ; the appearance of the child abuse industry and a group of professionals dedicated to the protection of children ; and the popularization of posttraumatic stress disorder, repressed memory and corresponding survivor movement.

Exokernel is an operating system kernel developed by the MIT Parallel and Distributed Operating Systems group, and also a class of similar operating systems.

that is surjective and its kernel is the special linear group.

of groups and group homomorphisms is called exact if the image ( or range ) of each homomorphism is equal to the kernel of the next:

The kernel of d ( i. e., the subgroup sent to 0 by d ) is the group of cycles, or in the case of a cochain complex, cocycles.

This is the normal subgroup of the general linear group given by the kernel of the determinant

This is a group homomorphism ; its kernel is the group of units of R, and its cokernel is the ideal class group of R. The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.

Since unit quaternions can be used to represent rotations in 3-dimensional space ( up to sign ), we have a surjective homomorphism from SU ( 2 ) to the rotation group SO ( 3 ) whose kernel is

The condition Ext < sup > 1 </ sup >( A, Z ) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f: B → A is a surjective group homomorphism whose kernel is isomorphic to the group of integers Z, then there exists a group homomorphism g: A → B with fg = id < sub > A </ sub >.

We define the kernel of h to be the set of elements in G which are mapped to the identity in H

The kernel of h is a normal subgroup of G ( in fact, h ( g < sup >- 1 </ sup > u g )

* There is some homomorphism on G for which N is the kernel.

# The kernel of φ is a normal subgroup of G,

* If G and H are groups and φ: G → H is a homomorphism, then the index of the kernel of φ in G is equal to the order of the image:

; indeed, N can be taken to be the kernel of the natural homomorphism from G to the permutation group of the left ( or right ) cosets of H.

In detail, the space of homomorphisms from G to the ( cyclic ) group of order p, is a vector space over the finite field A non-trivial such map has as kernel a normal subgroup of index p, and multiplying the map by an element of ( a non-zero number mod p ) does not change the kernel ; thus one obtains a map from

* For any positive integer n, the group μ < sub > n </ sub > is the kernel of the nth power map from G < sub > m </ sub > to itself.