In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism A → k {\displaystyle A\to k} , typically...

side-by-side Augmentation (geometry), a way of enlarging a polyhedron Augmentation (algebra), a certain algebra homomorphism Augmentation ideal, in mathematics...

In algebra, an augmentation ideal is an ideal that can be defined in any group ring. If G is a group and R a commutative ring, there is a ring homomorphism...

In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact...

In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group. Let R...

(with respect to the augmentation ideal) of its group ring QG, with inverse given by the group of grouplike elements of a Hopf algebra H, essentially those...

Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods...

dataset. In such cases, data augmentation can be applied, to allow training GAN on smaller datasets. Naïve data augmentation, however, brings its problems...

abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology...

In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec C = Spec X ⁡ R {\displaystyle...

functions or operations that can be applied to the data, i.e., it is an algebraic structure about data. Data structures serve as the basis for abstract...

Abu Kamil an important part in introducing algebra to Europe. Abu Kamil made important contributions to algebra and geometry. He was the first Islamic mathematician...

optimality" prescribes. The equation applies to algebraic structures with a total ordering; for algebraic structures with a partial ordering, the generic...

theorem. The Kurosh problem on group algebras concerns the augmentation ideal I. If I is a nil ideal, is the group algebra locally nilpotent? There is an important...

r=I\{s_{1},s_{2},..,s_{n}\}} is an algebraic function combining the subjective values of the information. "Cognitive algebra" refers to the class of functions...

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with...

{\displaystyle A[S]} . For an A {\displaystyle A} -algebra B {\displaystyle B} , this comes with a natural augmentation map η B : A [ B ] → B {\displaystyle \eta...

ordinary differential equations by the Runge–Kutta method. It arose from an algebraic formalism involving rooted trees that provides formal power series solutions...

theory (or Set of sets that do not contain themselves) σ-algebra – Algebraic structure of set algebra σ-ring – Family of sets closed under countable unions...

Matroid theory borrows extensively from the terms used in both linear algebra and graph theory, largely because it is the abstraction of various notions...

2011. Al-Sammarraie, Ahmed T.; Vafai, Kambiz (2017). "Heat transfer augmentation through convergence angles in a pipe". Numerical Heat Transfer, Part...

be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra. A collection...

In algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that two homology theories are isomorphic. The...

the outer face. The dual of this augmented planar graph is itself the augmentation of another st-planar graph. In a directed plane graph, the dual graph...

the field of rational numbers Q. Given the square-free integer c, the augmentation of Q by quadratic irrationals using √c produces a quadratic field Q(√c)...

1939 – 2 October 2021) was an Indian mathematician who specialised in algebra. IBS Passi was the former dean of university instructions (DUI) and professor...

the inhibitory response most of the time. They also noted a temporary augmentation of the excitatory response occurring after the attenuation. As a control...

In homological algebra, the Cartan–Eilenberg resolution is in a sense, a resolution of a chain complex. It can be used to construct hyper-derived functors...

\colon C_{0}\to \mathbb {Z} } denotes the ring homomorphism known as the augmentation map, which is defined as follows: if n 1 σ 1 + ⋯ + n k σ k ∈ C 0 {\displaystyle...

the G-equivariant K-theory of X with respect to I, where I denotes the augmentation ideal of the representation ring of G. In the special case of X being...