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Concept in abstract algebra
under the Frobenius endomorphism F*. Brion & Kumar (2005) give a detailed discussion of Frobenius splittings. A fundamental property of Frobenius-split projective
Frobenius_splitting
category Frobenius characteristic map Frobenius coin problem Frobenius number Frobenius companion matrix Frobenius covariant Frobenius element Frobenius endomorphism
List of things named after Ferdinand Georg Frobenius
List_of_things_named_after_Ferdinand_Georg_Frobenius
Indian mathematician
of Frobenius splitting of algebraic varieties jointly with Vikram Bhagvandas Mehta in (Mehta & Ramanathan 1985). The notion of Frobenius splitting led
Annamalai_Ramanathan
Describes statistically the splitting of primes in a given Galois extension of Q
primes. More generally, splitting behavior can be specified by assigning to (almost) every prime number an invariant, its Frobenius element, which is a representative
Chebotarev_density_theorem
Aspect of algebraic number theory
The theory of the Frobenius element goes further, to identify an element of DPj / IPj for given j which corresponds to the Frobenius automorphism in the
Splitting of prime ideals in Galois extensions
Splitting_of_prime_ideals_in_Galois_extensions
Type of ring in commutative algebra
related to Frobenius splitting: A Noetherian local ring A {\displaystyle A} of positive characteristic p is regular if and only if the Frobenius morphism
Regular_local_ring
1991 Vikram Bhagvandas Mehta Maharashtra Frobenius split 1991 Annamalai Ramanathan Tamil Nadu Frobenius splitting 1992 Maithili Sharan Rajasthan Mathematical
List of Shanti Swarup Bhatnagar Prize recipients
List_of_Shanti_Swarup_Bhatnagar_Prize_recipients
Deformation theory Differential graded Lie algebra Kodaira–Spencer map Frobenius splitting Relative effective Cartier divisor M. Artin, Lectures on Deformations
Degeneration (algebraic geometry)
Degeneration_(algebraic_geometry)
4.1.2 and 4.1.3 Smith, Karen E.; Zhang, Wenliang (2014-09-03). "Frobenius Splitting in Commutative Algebra". arXiv:1409.1169 [math.AC]. Grothendieck
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
ISSN 0021-8693, MR 1338967 Mehta, V. B.; Ramanathan, A. (1985), "Frobenius splitting and cohomology vanishing for Schubert varieties", Annals of Mathematics
Demazure_module
Indian mathematician
Kac-Moody groups, their flag varieties, and representation theory and Frobenius splitting methods in geometry and representation theory (jointly with Michel
Shrawan_Kumar_(mathematician)
In ring theory and Frobenius algebra extensions, areas of mathematics, there is a notion of depth two subring or depth of a Frobenius extension. The notion
Depth of noncommutative subrings
Depth_of_noncommutative_subrings
Nakayama, On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings, Nagoya Math. J. Volume 9 (1955), 1–16. Endo, Shizuo;
Separable_algebra
Algebraic structure
{\displaystyle \mathrm {GF} (p)} . It is called the Frobenius automorphism, after Ferdinand Georg Frobenius. Denoting by φk the composition of φ with itself
Finite_field
Indian mathematician
169, 1–39 (2009). (With V.B. Mehta) "Moduli of vector bundles, Frobenius splitting, and invariant theory", Ann. of Math. 144, 269–313 (1996). "Factorisation
T._R._Ramadas
Indian mathematician (1946–2014)
geometry. Bhagvandas Mehta, Vikram; Ramanathan, Annamalai (July 1985). "Frobenius splitting and cohomology vanishing for Schubert varieties". Annals of Mathematics
Vikram_Bhagvandas_Mehta
Matrix in mathematics
identity matrix. For the non-singularity of A, according to the Perron–Frobenius theorem, it must be the case that s > ρ(B). Also, for a non-singular M-matrix
M-matrix
Different ideal Dedekind domain Splitting of prime ideals in Galois extensions Decomposition group Inertia group Frobenius automorphism Chebotarev's density
List of algebraic number theory topics
List_of_algebraic_number_theory_topics
Representation theory of groups
general, such a structure is called a Frobenius algebra. As the name implies, these were introduced by Frobenius in the nineteenth century. They have been
Regular_representation
Transforms equations for numerical solution
\|_{F}} is the Frobenius norm and T = P − 1 {\displaystyle T=P^{-1}} is from some suitably constrained set of sparse matrices. Under the Frobenius norm, this
Preconditioner
Square matrix constructed from a monic polynomial
In linear algebra, the Frobenius companion matrix of the monic polynomial p ( x ) = c 0 + c 1 x + ⋯ + c n − 1 x n − 1 + x n {\displaystyle p(x)=c_{0}+c_{1}x+\cdots
Companion_matrix
Dutch mathematician
over 240 citations. van der Kallen, Wilberd (1993). Lectures on Frobenius Splittings and B-modules. ISBN 978-81-85198-60-6; 98 pages{{cite book}}: CS1
Wilberd_van_der_Kallen
Sum of elements on the main diagonal
B is a square matrix. The Frobenius inner product and norm arise frequently in matrix calculus and statistics. The Frobenius inner product may be extended
Trace_(linear_algebra)
Snake lemma Splitting lemma Yoneda lemma Matrix determinant lemma Matrix inversion lemma Burnside's lemma also known as the Cauchy–Frobenius lemma Frattini's
List_of_lemmas
Mathematical group
\operatorname {Gal} (E/F)} is cyclic of order n and generated by the Frobenius homomorphism. The field extension Q ( 2 , 3 ) / Q {\displaystyle \mathbb
Galois_group
Elementwise product of two matrices
artificial neural network models, specifically convolutional layers. Frobenius inner product Pointwise product Kronecker product Khatri–Rao product Horn
Hadamard_product_(matrices)
Mathematical space
non-degeneracy' condition called 'complete non-integrability'. From the Frobenius theorem, one recognizes the condition as the opposite of the condition
3-manifold
Polynomial coprime with its derivative
2) if P is irreducible, but most resolvents are not always separable. Frobenius endomorphism Pages 240-241 of Lang, Serge (1993), Algebra (Third ed.)
Separable_polynomial
{\displaystyle 1<\rho (T_{J})<\rho (T_{1})} . The proof uses the Perron-Frobenius theorem for non-negative matrices. Its proof can be found in Richard S
Stein-Rosenberg_theorem
Field theory theorem
{\displaystyle \alpha =g(T,U)} in E ∖ F {\displaystyle E\setminus F} , the Frobenius endomorphism shows that the element α p {\displaystyle \alpha ^{p}} lies
Primitive_element_theorem
Branch of mathematics that studies algebraic structures
domain, Dedekind domain, Prüfer domain Von Neumann regular ring Quasi-Frobenius ring Hereditary ring, Semihereditary ring Local ring, Semi-local ring
List of abstract algebra topics
List_of_abstract_algebra_topics
theory) Focal subgroup theorem (abstract algebra) Frobenius determinant theorem (group theory) Frobenius reciprocity theorem (group representations) Frucht's
List_of_theorems
Concept in differential geometry
each point are smooth distributions which are integrable in the sense of Frobenius. The integral manifolds of these distributions are totally geodesic submanifolds
Holonomy
Field extension of the rational numbers by a primitive root of unity
{\displaystyle q} is a prime not dividing n {\displaystyle n} , then the Frobenius element Frob q ∈ Gal ( Q ( ζ n ) / Q ) {\displaystyle \operatorname
Cyclotomic_field
Operation in group theory
the existence of a decomposition as a semidirect product (also known as splitting extension). Given a group G with identity element e, a subgroup H, and
Semidirect_product
Group of unitary matrices
\alpha \colon x\mapsto x^{q}} (the r {\displaystyle r} th power of the Frobenius automorphism). This allows one to define a Hermitian form on an F q 2
Unitary_group
Subfield of convex optimization
the maximum Frobenius norm of a feasible solution, and ε>0 a constant. A matrix X in Sn is called ε-deep if every matrix Y in L with Frobenius distance at
Semidefinite_programming
Non-abelian group of order eight
_{9}\\\mu _{z}(a+bk)=z\cdot (a+bk)\end{cases}}} In addition we have the Frobenius automorphism ϕ ( a + b k ) = ( a + b k ) 3 {\displaystyle \phi (a+bk)=(a+bk)^{3}}
Quaternion_group
Theory in abstract algebra
field k of positive characteristic p, G is the Galois group, π is the Frobenius map minus the identity, and C the finite field of order p. Taking A to
Kummer_theory
Bound on eigenvalues
is satisfied here. For matrices with non-negative entries, see Perron–Frobenius theorem. Doubly stochastic matrix Hurwitz-stable matrix – Matrix whose
Gershgorin_circle_theorem
subgroup. The relation between restriction and induction is described by Frobenius reciprocity and the Mackey theorem. Restriction to a normal subgroup behaves
Restricted_representation
Type of Dirichlet series associated to number field extensions
vital features of most number fields. The Frobenius element need not be unique. For unramified primes all Frobenius elements are conjugates in the Galois
Artin_L-function
Statement in abstract algebra
various canonical forms: invariant factors + companion matrix yields Frobenius normal form (aka, rational canonical form) primary decomposition + companion
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
Mathematical operation in linear algebra
include: Block matrix operations Cracovian product, defined as A ∧ B = BTA Frobenius inner product, the dot product of matrices considered as vectors, or,
Matrix_multiplication
Algebraic structure with addition, multiplication, and division
pth power, i.e., the p-fold product of the element a. Therefore, the Frobenius map F → F : x ↦ xp is compatible with the addition in F (and also with
Field_(mathematics)
Submodule of a mathematical ring
Boolean prime ideal theorem Ideal theory Ideal (order theory) Ideal norm Splitting of prime ideals in Galois extensions Ideal sheaf Some authors call the
Ideal_(ring_theory)
Mathematical concept named for Ernst Witt
operator F {\displaystyle F} which is called the Frobenius operator since it reduces to the Frobenius operator on k {\displaystyle k} . Witt observed that
Witt_vector
Use of mathematical groups in magnetochemistry
finite subgroups of SU(2) and SO(3) were determined and tabulated by F. G. Frobenius in 1898, with alternative derivations by I. Schur and H. E. Jordan in
Finite_subgroups_of_SU(2)
Group with subnormal series where all factors are abelian
− 1 {\displaystyle a\in F_{i-1}} F m {\displaystyle F_{m}} contains a splitting field for f ( x ) {\displaystyle f(x)} The smallest Galois field extension
Solvable_group
Set with associative invertible operation
groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups
Group_(mathematics)
Field theory is the branch of algebra that studies fields
field A field over which no biquaternion algebra is a division algebra. Frobenius field A pseudo algebraically closed field whose absolute Galois group
Glossary_of_field_theory
Mythical female creature
to her father's house. In a Kabylian tale collected by ethnologist Leo Frobenius, titled Die Taubenfrauen ("The Dove Maidens"), a young hunter journeys
Swan_maiden
In mathematics, a partition of a manifold into submanifolds
a codimension n − 1 foliation). This observation generalises to the Frobenius theorem, saying that the necessary and sufficient conditions for a distribution
Foliation
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
pseudospectra are better alternatives. Canonical basis Canonical form Frobenius normal form Jordan matrix Jordan–Chevalley decomposition Matrix decomposition
Jordan_normal_form
subsequently flushed by the down-sampling. You can avoid their computation by splitting the filters and the signal into even and odd indexed values before the
Polyphase_matrix
Book about number theory
extension of A-fields, then any automorphism of K over k is induced by the Frobenius automorphism for infinitely many places of K. This approach also allows
Basic_Number_Theory
Mathematical structure in differential geometry
distribution; it is easy to check that it is involutive, therefore, by the Frobenius theorem, M {\displaystyle M} admits a partition into leaves. Moreover
Poisson_manifold
Process in machine learning and statistics
\ldots ,x_{n}\geq 0,} where ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{F}} is the Frobenius norm. The optimization problem is a Lasso problem, and thus it can be
Feature_selection
Algebraic construct of interest in theoretical physics
solvable rather than semisimple Lie groups. They are associated to Lie splittings of Lie algebras or local factorisations of Lie groups and can be viewed
Quantum_group
Mathematical object
{\displaystyle [1_{K},\chi \downarrow _{K}^{G}]=[1\uparrow _{K}^{G},\chi ]} by Frobenius reciprocity and 1 ↑ K G {\displaystyle 1\uparrow _{K}^{G}} is the character
Gelfand_pair
Mathematical group of loops in a Lie group
closed normal subgroup of LG. The inclusion of constant loops gives a splitting of ev1, so there is a split exact sequence 1 → Ω G → L G → ev 1 G → 1
Loop_group
Double cover Lie group of the special orthogonal group
sequence, and one has a sequence of three groups, Spin(n) → SO(n) → PSO(n), splitting by parity yields: Spin(2n) → SO(2n) → PSO(2n), Spin(2n+1) → SO(2n+1) = PSO(2n+1)
Spin_group
Study of vector bundles, principal bundles, and fibre bundles
{\mathfrak {g}})} corresponding to the curvature. From the perspective of the Frobenius integrability theorem, the curvature measures precisely the extent to
Gauge_theory_(mathematics)
Best results achieved to date
Antoine; Pierrot, Cécile. "Improving the Polynomial time Precomputation of Frobenius Representation Discrete Logarithm Algorithms" (PDF). Archived from the
Discrete_logarithm_records
Number used to approximate the square root of 2
integer solution to a2 + b2 = c2 where a + 1 = b. The next table shows that splitting the odd number Hn into nearly equal halves gives a square triangular number
Pell_number
Number of close-packed spheres in an octahedron
partitioned into two square pyramids, one upside-down underneath the other, by splitting it along a square cross-section. Therefore, the n {\displaystyle n} th
Octahedral_number
Infinitesimal version of Lie groupoid
injective anchor (hence foliation algebroids) are always integrable (by Frobenius theorem) Lie algebra bundle are always integrable Action Lie algebroids
Lie_algebroid
Clans in western Africa
English) Universität Frankfurt am Main, Frobenius-Institut, Deutsche Gesellschaft für Kulturmorphologie, Frobenius Gesellschaft, "Paideuma: Mitteilungen
Serer_maternal_clans
FROBENIUS SPLITTING
FROBENIUS SPLITTING
FROBENIUS SPLITTING
Boy/Male
Muslim/Islamic
Sun of religion
Boy/Male
Muslim/Islamic
Flame blaze
Boy/Male
Hindu, Indian, Marathi
Swift; Hero; Strong
Boy/Male
Hindu, Indian
Atom
Girl/Female
Italian American Gaelic Latin Shakespearean
Blesses.
Surname or Lastname
English
English : unexplained.Americanized spelling of German Eimes, a patronymic from a short form of the Germanic personal name Agimo, formed with agi ‘point (of a sword or lance)’ (Old High German ecka).
Boy/Male
German
From the Linden Tree Hill
Male
Native American
Native American Cheyenne name HESKOVIZENAKO means "porcupine bear."
Boy/Male
Tamil
Lord Shiva
Boy/Male
Hindu, Indian, Malayalam, Marathi
Actress
FROBENIUS SPLITTING
FROBENIUS SPLITTING
FROBENIUS SPLITTING
FROBENIUS SPLITTING
FROBENIUS SPLITTING
n.
A split of a sheepskin; one of the thin sections made by splitting a sheepskin with a cutting knife or machine.
n.
A fragment or part of anything separated from the whole, in any manner, as by cutting, splitting, breaking, or tearing; a part; a portion; as, a piece of sugar; to break in pieces.
n.
A pen for writing made by sharpening and splitting the point or nib of the stock of a feather; as, history is the proper subject of his quill.
n.
The cutting tool or machine used in splitting leather or skins, as sheepskins.
n.
One of the strips at the end of a bandage formed by splitting the bandage one or more times.
v. t.
To cure, by splitting, salting, and smoking.
n.
A cleaving, splitting, or breaking up into parts.
n.
The act of paring or splitting leather or skins.
n.
A wooden wedge used in splitting blocks.
a.
Resembling slate; having the nature, appearance, or properties, of slate; composed of thin parallel plates, capable of being separated by splitting; as, a slaty color or texture.
n.
A tool for splitting wood into shingles; a frow.
n.
A piece made in paring or splitting leather; specifically, the part from the inner, or flesh, side.
a.
Pertaining to, or containing, mica; splitting into laminae or leaves like mica.
n.
A genus of budding fungi, the various species of which have the power, to a greater or less extent, or splitting up sugar into alcohol and carbonic acid. They are the active agents in producing fermentation of wine, beer, etc. Saccharomyces cerevisiae is the yeast of sedimentary beer. Also called Torula.
p. pr. & vb. n.
of Split
n.
A piece of metal, or other hard material, thick at one end, and tapering to a thin edge at the other, used in splitting wood, rocks, etc., in raising heavy bodies, and the like. It is one of the six elementary machines called the mechanical powers. See Illust. of Mechanical powers, under Mechanical.
n.
An opening made by riving or splitting; a cleft; a fissure.
a.
Deafening; disagreeably loud or shrill; as, ear-splitting strains.
n.
A piece that is split off, or made thin, by splitting; a splinter; a fragment.