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Equivalence relation expressing that two elements have the same image under a function
In set theory, the kernel of a function f {\displaystyle f} (or equivalence kernel) may be taken to be either the equivalence relation on the function's
Kernel_(set_theory)
Topics referred to by the same term
to the zero vector Kernel (category theory), a generalization of the kernel of a homomorphism Kernel (set theory), an equivalence relation: partition
Kernel
Mathematical concept
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation,
Poisson_kernel
Concept in probability theory
In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes
Markov_kernel
Class of algorithms for pattern analysis
theory (for example, using Rademacher complexity). Kernel methods can be thought of as instance-based learners: rather than learning some fixed set of
Kernel_method
Elements taken to zero by a homomorphism
respective kernel would be the even integers which all have 0 as its parity. The kernel of a homomorphism of group-like structures will be a singleton set that
Kernel_(algebra)
Set of arguments where two or more functions have the same value
equation. In certain contexts, a difference kernel is the equaliser of exactly two functions. Let X and Y be sets. Let f and g be functions, both from X to
Equaliser_(mathematics)
Generalization of a positive-definite matrix
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix
Positive-definite_kernel
Mathematical theorem
in the reproducing kernel Hilbert space theory where it characterizes a symmetric positive-definite kernel as a reproducing kernel. To explain Mercer's
Mercer's_theorem
Mathematical function
mathematics of probability, a transition kernel or kernel is a function in mathematics that has different applications. Kernels can for example be used to define
Transition_kernel
Algorithmic technique
parameterized complexity theory, it is often possible to prove that a kernel with guaranteed bounds on the size of a kernel (as a function of some parameter
Kernelization
In functional analysis, a Hilbert space
In functional analysis, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Core of a computer operating system
kernel is a computer program at the core of a computer's operating system that always has complete control over everything in the system. The kernel is
Kernel_(operating_system)
In mathematics, the Carathéodory kernel theorem is a result in complex analysis and geometric function theory established by the Greek mathematician Constantin
Carathéodory_kernel_theorem
Game where groups of players may enforce cooperative behaviour
much collective payoff a set of players can gain by forming a coalition. Cooperative game theory is a branch of game theory that deals with the study
Cooperative_game_theory
Concept in statistics
In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method
Kernel_density_estimation
Mathematical theory of integral equations
abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space. It therefore
Fredholm_theory
A family of simple undirected graphs defined by spectral properties
In graph theory, a nut graph is a finite simple graph with at least two vertices whose adjacency matrix has nullity one and whose kernel is spanned by
Nut_graph_(graph_theory)
Operating system microkernel
Mach (/mɑːk/) is an operating system kernel developed at Carnegie Mellon University by Richard Rashid and Avie Tevanian to support operating system research
Mach_(kernel)
Family of subsets representing "large" sets
topology, including set theory, mathematical logic, model theory (ultraproducts for example), abstract algebra, and others. Filters on a set were later generalized
Filter_on_a_set
Type of kernel induced by artificial neural networks
study of artificial neural networks (ANNs), the neural tangent kernel (NTK) is a kernel that describes the evolution of deep artificial neural networks
Neural_tangent_kernel
Statistical technique
defined by the kernel, such that closer points are given higher weights. The estimated function is smooth, and the level of smoothness is set by a single
Kernel_smoother
Overview of and topical guide to machine learning
analytics VC dimension VIGRA Validation set Vapnik–Chervonenkis theory Variable-order Bayesian network Variable kernel density estimation Variable rules analysis
Outline_of_machine_learning
(mutational analysis). The basic problem of viability theory is to find the "viability kernel" of an environment, the subset of initial states in the
Viability_theory
Set of methods for supervised statistical learning
using the kernel trick, representing the data only through a set of pairwise similarity comparisons between the original data points using a kernel function
Support_vector_machine
Class of nonparametric methods
manifolds, dynamical systems, and other structured objects. The theory behind kernel embeddings of distributions has been primarily developed by Alex
Kernel embedding of distributions
Kernel_embedding_of_distributions
Transformations induced by a mathematical group
set of points x ∈ X {\displaystyle x\in X} such that the map g ↦ g ⋅ x {\displaystyle g\mapsto g\cdot x} is smooth. There is a well-developed theory of
Group_action
Mathematical category whose hom sets form Abelian groups
without kernels and/or cokernels. There is a convenient relationship between the kernel and cokernel and the abelian group structure on the hom-sets. Given
Preadditive_category
Branch of mathematics that studies the properties of groups
Given any set F of generators { g i } i ∈ I {\displaystyle \{g_{i}\}_{i\in I}} , the free group generated by F surjects onto the group G. The kernel of this
Group_theory
Category
mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail
Pre-abelian_category
Statistical learning theory
reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data
Representer_theorem
Model for approximating non-linear effects, similar to a Taylor series
Bernhard Schölkopf (2006). "A unifying view of Wiener and Volterra theory and polynomial kernel regression". Neural Computation. 18 (12): 3097–3118. doi:10.1162/neco
Volterra_series
Concept in mathematics
Frobenius kernel K. (This is a theorem due to Frobenius (1901); there is still no proof of this theorem that does not use character theory, although see
Frobenius_group
Type of group in abstract algebra
and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is
Symmetric_group
topologically sufficiently rich enough, then the Markov operator admits a kernel representation. Markov operators can be linear or non-linear. Closely related
Markov_operator
Category whose objects are measurable spaces and whose morphisms are Markov kernels
category of Markov kernels, often denoted Stoch, is the category whose objects are measurable spaces and whose morphisms are Markov kernels. It is analogous
Category_of_Markov_kernels
Bayesian interpretation of kernel regularization examines how kernel methods in machine learning can be understood through the lens of Bayesian statistics
Bayesian interpretation of kernel regularization
Bayesian_interpretation_of_kernel_regularization
Mathematical object that generalizes the standard notions of sets and functions
areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations
Category_(mathematics)
Map (arrow) between two objects of a category
category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous
Morphism
Operating system kernel designed as a replacement for Unix
replacement for the Unix kernel, and released as free software under the GNU General Public License. When the Linux kernel proved to be a viable solution
GNU_Hurd
Group of even permutations of a finite set
(14)(23) }, that is the kernel of the surjection of A4 onto A3 ≅ Z3. We have the exact sequence V → A4 → A3 = Z3. In Galois theory, this map, or rather the
Alternating_group
Framework for machine learning
statistical learning theory, supervised learning is best understood. Supervised learning involves learning from a training set of data. Every point in
Statistical_learning_theory
Sets whose elements have degrees of membership
kernel Kern ( A ) {\displaystyle \operatorname {Kern} (A)} ). Note that some authors understand "kernel" in a different way; see below. A fuzzy set
Fuzzy_set
Quotient space of a codomain of a linear map by the map's image
called the corank of f. Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to the domain)
Cokernel
Mathematical set with an ordering
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The
Partially_ordered_set
Subgroup invariant under conjugation
Furthermore, the normal subgroups of G {\displaystyle G} are precisely the kernels of group homomorphisms with domain G , {\displaystyle G,} which means that
Normal_subgroup
Relationship between two functors abstracting many common constructions
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence
Adjoint_functors
Theorem
In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It
Schwartz_kernel_theorem
It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure. Let G {\displaystyle G}
Positive-definite function on a group
Positive-definite_function_on_a_group
Mathematical construction used in homotopy theory
Simplicial sets are used to define quasi-categories, a basic notion of higher category theory. A construction analogous to that of simplicial sets can be
Simplicial_set
Mathematical category with finite limits and coequalizers
In category theory, a regular category is a category with finite limits and coequalizers of all pairs of morphisms called kernel pairs, satisfying certain
Regular_category
Mathematical group that can be generated as the set of powers of a single element
p-adic numbers), that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it
Cyclic_group
element eA of A must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set {eA}. The notion of ideal generalises
Malcev_algebra
Category with direct sums and certain types of kernels and cokernels
is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical
Abelian_category
Boot loader package
multiple operating systems installed on a computer set up for multi-booting or select a specific kernel configuration available on a particular operating
GNU_GRUB
Overview of and topical guide to category theory
functor Yoneda lemma Product (category theory) Equaliser (mathematics) Kernel (category theory) Pullback (category theory)/fiber product Inverse limit Pro-finite
Outline_of_category_theory
surjective and has kernel B. This quotient category can be constructed as a localization of A by the class of morphisms whose kernel and cokernel are both
Localization_of_a_category
with a specific general theory of software engineering. This theory should be solidly based on the SEMAT Essence language and kernel, and should support software
SEMAT
Category whose objects are groups and whose morphisms are group homomorphisms
{\displaystyle \mathbf {Grp} } has a category-theoretic kernel (given by the ordinary kernel of algebra ker f = { x ∈ G | f ( x ) = e } {\displaystyle
Category_of_groups
Mathematical concept for comparing objects
only if }}f(x)=f(y).} The equivalence kernel of an injection is the identity relation. A partition of X is a set P of nonempty subsets of X, such that
Equivalence_relation
classification, the Fisher kernel, named after Ronald Fisher, is a function that measures the similarity of two objects on the basis of sets of measurements for
Fisher_kernel
General theory of mathematical structures
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the
Category_theory
Mathematical concept
space with the hull-kernel topology (or Jacobson topology). This is defined as follows: If X is a set of primitive ideals, its hull-kernel closure is X ¯ =
Spectrum_of_a_C*-algebra
Number in {..., –2, –1, 0, 1, 2, ...}
the set of the integers was not used before the end of the 19th century, when Georg Cantor introduced the concept of infinite sets and set theory. The
Integer
integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied
Fredholm_integral_equation
French mathematician (1915–2002)
mathematician who received the Fields Medal in 1950 for pioneering the theory of distributions or generalized functions, giving a well-defined meaning
Laurent_Schwartz
Monster and modular connection
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular
Monstrous_moonshine
Generalized function whose value is zero everywhere except at zero
represented by integration against a kernel K z ( ζ ) {\displaystyle K_{z}(\zeta )} , the Bergman kernel. This kernel is the analog of the delta function
Dirac_delta_function
Smallest normal group containing a set
In group theory, the normal closure of a subset S {\displaystyle S} of a group G {\displaystyle G} is the smallest normal subgroup of G {\displaystyle
Normal_closure_(group_theory)
Kernel that provides fewer services than a traditional kernel
In computer science, a microkernel (often abbreviated as μ-kernel) is the near-minimum amount of software that can provide the mechanisms needed to implement
Microkernel
Algebraic curve in mathematics
proven with the help of some general theory; see local zeta function and étale cohomology for example. The set of points E(Fq) is a finite abelian group
Elliptic_curve
Concept in mathematical group theory
is simple) can be recognised from its character table. The kernel of a character χ is the set of elements g in G for which χ(g) = χ(1); this is a normal
Character_theory
Periodic set of points
In geometry and group theory, a lattice in the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} is an infinite set of points in this space with
Lattice_(group)
Set of the values of a function
Fiber (mathematics) – Set of all points in a function's domain that all map to some single given point Image (category theory) Kernel of a function – Equivalence
Image_(mathematics)
Mathematical category
results of topos theory. The category of sets is an important special case: it plays the role of a point in topos theory. Indeed, a set may be thought of
Topos
Theoretical framework in harmonic analysis
characteristic function of a set used in the definition of fρ can be replaced by a smoother function. A key estimate of Littlewood–Paley theory is the Littlewood–Paley
Littlewood–Paley_theory
Commutative group (mathematics)
whose inverse is also an integer matrix). Changing the generating set of the kernel of M is equivalent with multiplying M on the right by a unimodular
Abelian_group
universe Institut Montpelliérain Alexander Grothendieck Serre–Grothendieck–Verdier duality Tarski–Grothendieck set theory Tutte–Grothendieck invariant
List of things named after Alexander Grothendieck
List_of_things_named_after_Alexander_Grothendieck
Abstract structure modeling spaces of probability measures
in probability theory whenever one considers probability measures which depend measurably on a parameter (giving rise to Markov kernels), or when one has
Giry_monad
Subgraph
itself a dominating set. One way to construct a fixed-parameter tractable algorithm for the nonblocker problem is to use kernelization, an algorithmic design
Nonblocker
Term in the mathematical area of order theory
mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop
Duality_(order_theory)
Mathematical function between groups that preserves multiplication structure
multiplication with −1; it is isomorphic to (Z/2Z, +). We define the kernel of h to be the set of elements in G that are mapped to the identity in H ker (
Group_homomorphism
Tasks in machine learning
data sets. In particular, three data sets are commonly used in different stages of the creation of the model: training, validation, and testing sets. The
Training, validation, and test data sets
Training,_validation,_and_test_data_sets
Mathematics concept
{\displaystyle G} is called finitely generated. The kernel ker φ {\displaystyle \ker \varphi } is the set of all relations in the presentation of G {\displaystyle
Free_group
Conditional independence of exchangeable observations
well as the Markov kernel X N → X N {\displaystyle X^{\mathbb {N} }\to X^{\mathbb {N} }} induced by it. In terms of category theory, we have a diagram
De_Finetti's_theorem
Generalization of category theory
In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows
Higher_category_theory
Mathematical structure with multiplication as its operation
In mathematics and group theory, the term multiplicative group refers to one of the following concepts: the group under multiplication of the invertible
Multiplicative_group
The set k X {\displaystyle k_{X}} , of all x ∈ M {\displaystyle x\in M} with p X ( x ) = 1 {\displaystyle p_{X}(x)=1} is called the kernel, the set of
Random_compact_set
Submodule of a mathematical ring
ideal as its kernel. Conversely, the kernel of a ring homomorphism is a two-sided ideal. Therefore, the two-sided ideals are exactly the kernels of ring homomorphisms
Ideal_(ring_theory)
continuous theory while either preserving or well approximating the desirable theoretical properties that lead to the choice of the Gaussian kernel (see the
Scale_space_implementation
Property in general topology
of a family of sets is called its kernel. Families with empty kernel are called free; those with nonempty kernel, fixed. The empty set cannot belong to
Finite_intersection_property
Tree-based ensemble machine learning methods
Erwan (2015). "Random forests and kernel methods". arXiv:1502.03836 [math.ST]. Breiman, Leo (2000). "Some infinity theory for predictor ensembles". Technical
Random_forest
Collection of sets in which every two sets have the same intersection
which the intersection of any two distinct sets is the same. This common intersection is called the kernel of the sunflower. The naming arises from a
Sunflower_(mathematics)
Inclusion of one mathematical structure in another, preserving properties of interest
R^{A}} . In model theory there is also a stronger notion of elementary embedding. In order theory, an embedding of partially ordered sets is a function F
Embedding
Theorem in category theory
theory, greatly simplifying the basic developments. Linton's monadicity theorem: Let A {\displaystyle {\mathcal {A}}} be a category that has kernel pairs
Beck's_monadicity_theorem
Particular correspondence between two partially ordered sets
especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections
Galois_connection
Group obtained by aggregating similar elements of a larger group
{\displaystyle G} (since it is the kernel of the determinant homomorphism). The cosets of N {\displaystyle N} are the sets of matrices with a given determinant
Quotient_group
Yang–Mills theory in two dimensions with a well-defined measure
retrospect, be seen to be connected to the heat kernel on the structure group of the theory. The role of the heat kernel was made more explicit in various works
Two-dimensional Yang–Mills theory
Two-dimensional_Yang–Mills_theory
Generalized object in category theory
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas
Product_(category_theory)
Analyzes the topology of a manifold by studying differentiable functions on that manifold
manifold whose critical set is a closed submanifold and whose Hessian is non-degenerate in the normal direction. (Equivalently, the kernel of the Hessian at
Morse_theory
KERNEL SET-THEORY
KERNEL SET-THEORY
Girl/Female
American, Australian, British, Christian, English, Gaelic, German, Greek, Hebrew, Irish, Latin
Shining Sea; Sea-bright; Bitterness; Girl who Shines Like the Sea
Male
Romanian
Romanian form of Greek Kornelios, CORNEL means "of a horn."
Girl/Female
Australian, Chinese, Christian, Danish, German, Irish
Kernel; Nut
Female
Egyptian
, an uncertain goddess.
Girl/Female
Australian, Celtic, Christian, Irish
Graceful; Kernel
Male
Scandinavian
Scandinavian form of English Kenneth, KENNET means both "comely; finely made" and "born of fire."Â
Male
Scandinavian
Scandinavian form of German Werner, VERNER means "Warin warrior," i.e. "covered warrior."
Female
Hebrew
(כַּרְמֶל) Hebrew unisex name KARMEL means "garden-land." In the bible, this is the name of a mountain in the Holy Land.
Female
English
Medieval English contracted form of Roman Latin Petronel, PERONEL means "little rock."
Male
Slovene
Slovene form of Greek Bartholomaios, JERNEJ means "son of Talmai."
Boy/Male
French
Akernel.
Male
Polish
Polish form of Roman Latin Cornelius, KORNELI means "of a horn."
Male
English
Middle English form of Anglo-Saxon Cenhelm, KENELM means "keen protection."Â
Surname or Lastname
English
English : variant spelling of Pennell (see Parnell).
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Boy/Male
Egyptian Hebrew Swedish
Son of Seb and Nut.
Female
English
Variant form of English Keren, KERENA means "horn (of an animal)."Â
Female
English
Variant spelling of English Muriel, MERIEL means "sea-bright."
Surname or Lastname
Swedish
Swedish : ornamental name formed with the common surname suffix -ell. The first element is unexplained, possibly from a place-name.English, Scottish, and northern Irish : unexplained; possibly a respelling of Scottish Kerneil, a habitational name from Carneil in Carnock, Fife.
Girl/Female
Australian, Celtic, Christian, Irish
Kernel; Nut
KERNEL SET-THEORY
KERNEL SET-THEORY
Boy/Male
Hindu
Bowed down, Modest, To bow in a humble greeting
Boy/Male
Bengali, Hindu, Indian, Jain, Marathi
Peace; To have Control; Patience
Girl/Female
Sikh
Lords glory
Girl/Female
Sikh
Metrical composition
Girl/Female
Indian, Tamil
Pet Parrot
Girl/Female
Hindu, Indian
Prayer of Sai Baba
Boy/Male
English American French
Girl/Female
Gujarati, Hindu, Indian, Kannada, Marathi, Telugu
Beautiful
Boy/Male
Muslim
Character of a person, Heart, Mind, Conscience
Male
Slavic
(ВелеÑÑŠ) Variant form of Slavic Volos, VELES means "ox." In mythology, this is the name of a god of the earth, underworld, dragons, cattle, magic and trickery. He is an enemy of Perun and is described as being horned and serpentine.Â
KERNEL SET-THEORY
KERNEL SET-THEORY
KERNEL SET-THEORY
KERNEL SET-THEORY
KERNEL SET-THEORY
n.
See Weanel.
n.
A single seed or grain; as, a kernel of corn.
v. t.
To form with a kern. See 2d Kern.
a.
Of or pertaining to the spring; appearing in the spring; as, vernal bloom.
n.
The essential part of a seed; all that is within the seed walls; the edible substance contained in the shell of a nut; hence, anything included in a shell, husk, or integument; as, the kernel of a nut. See Illust. of Endocarp.
v. t.
To put or keep in a kennel.
imp. & p. p.
of Kern
n.
A small European evergreen oak (Quercus coccifera) on which the kermes insect (Coccus ilicis) feeds.
imp. & p. p.
of Kernel
n.
Any species of the genus Cornus, as C. florida, the flowering cornel; C. stolonifera, the osier cornel; C. Canadensis, the dwarf cornel, or bunchberry.
p. pr. & vb. n.
of Kernel
n.
The central, substantial or essential part of anything; the gist; the core; as, the kernel of an argument.
n.
See Kimnel.
n.
See Pimpernel.
a.
Having a kernel.
v. i.
To take the form of kernels; to granulate.
v. i.
To harden or ripen into kernels; to produce kernels.
a.
Full of kernels; resembling kernels; of the nature of kernels.