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Characteristic class of oriented, real vector bundles
algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted"
Euler_class
Topological invariant in mathematics
algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant
Euler_characteristic
mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Characteristic classes of vector bundles
Chern class is the first Chern class, which is an element of the second cohomology group of X. As it is the top Chern class, it equals the Euler class of
Chern_class
Association of cohomology classes to principal bundles
numbers, Pontryagin numbers, and the Euler characteristic. Given an oriented manifold M of dimension n with fundamental class [ M ] ∈ H n ( M ) {\displaystyle
Characteristic_class
Long exact sequence
cohomology class e called the Euler class of the bundle. Discussion of the sequence is clearest with de Rham cohomology. There cohomology classes are represented
Gysin_homomorphism
Ties Euler characteristic of a closed even-dimensional Riemannian manifold to curvature
( M ) {\displaystyle \chi (M)} denotes the Euler characteristic of M {\displaystyle M} . The Euler class is defined as e ( Ω ) = 1 ( 2 π ) n Pf ( Ω
Chern–Gauss–Bonnet_theorem
Concept in geometry
derivative is the virtual dimension. The localized Euler class of the pair (E,s) is a homology class with closed support on the zero set of the section
Localized_Chern_class
Continuous surjection satisfying a local triviality condition
bundle is partially characterized by its Euler class, which is a degree n + 1 {\displaystyle n+1} cohomology class in the total space of the bundle. In the
Fiber_bundle
Mathematical result in differential geometry
manifold with non-zero Euler class e ( T X ) {\displaystyle e(TX)} , then applying the Thom isomorphism and dividing by the Euler class, the topological index
Atiyah–Singer_index_theorem
Branch of algebraic geometry
) {\displaystyle \mathbb {G} (1,3)} . In order to get the Euler class, the total Chern class of T ∗ {\displaystyle T^{*}} must be computed, which is given
Schubert_calculus
American mathematician
compact surfaces His doctoral dissertation, "Discontinuous groups and the Euler class" (supervised by Morris W. Hirsch), characterizes discrete embeddings
William Goldman (mathematician)
William_Goldman_(mathematician)
Theorem on modular exponentiation
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers
Euler's_theorem
Algebraic structure used in topology
space X determines a cohomology class on X, the Euler class χ(E) ∈ Hr(X,Z). Informally, the Euler class is the class of the zero set of a general section
Cohomology
Generalization of an orientation of a vector space
invariant of an oriented bundle is the Euler class. The multiplication (that is, cup product) by the Euler class of an oriented bundle gives rise to a
Orientation of a vector bundle
Orientation_of_a_vector_bundle
vector bundle is a Chern class. A complex vector bundle is canonically oriented; in particular, one can take its Euler class. A complex vector bundle
Complex_vector_bundle
Difference between logarithm and harmonic series
\ln(x)} or log e ( x ) {\displaystyle \log _{e}(x)} . Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually
Euler's_constant
Cuboid whose edges and face diagonals have integer lengths
an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick
Euler_brick
Mathematical concept
resemble the Euler factors of an Euler product. Euler systems can be used to construct annihilators of ideal class groups or Selmer groups, thus giving
Euler_system
Trail in a graph that visits each edge once
posthumously in 1873 by Carl Hierholzer. This is known as Euler's Theorem: A connected graph has an Euler cycle if and only if every vertex has an even number
Eulerian_path
Graphical set representation involving overlapping shapes
An Euler diagram (/ˈɔɪlər/, OY-lər) is a diagrammatic means of representing sets and their relationships. They are particularly useful for explaining
Euler_diagram
Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular
Euler equations (fluid dynamics)
Euler_equations_(fluid_dynamics)
1974 textbook by John Milnor and James Stasheff
basic characteristic classes is described, which includes Stiefel–Whitney classes, Chern classes, Pontrjagin classes and the Euler class. Afterwards, it descibres
Characteristic_Classes
Integers occurring in the coefficients of the Taylor series of 1/cosh t
In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion 1 cosh t = 2 e
Euler_numbers
Set of topological invariants
(TS^{n})=2\neq 0} (where [Sn] denotes a fundamental class of Sn and χ the Euler characteristic), hence its Euler class e {\displaystyle e} must be nonzero. If we
Stiefel–Whitney_class
Characteristic class for real vector bundles
e(E)} denotes the Euler class of E {\displaystyle E} , and ⌣ {\displaystyle \smile } denotes the cup product of cohomology classes. As was shown by Shiing-Shen
Pontryagin_class
Diagram that shows all possible logical relations between a collection of sets
as by Christian Weise in 1712 (Nucleus Logicoe Wiesianoe) and Leonhard Euler in 1768 (Letters to a German Princess). The idea was popularised by Venn
Venn_diagram
Introduction and Reference on Differential Geometry
of characteristic classes of principal bundles (Chern–Weil theory), it covers Euler classes, Chern classes, and Pontryagin classes. The second volume
Foundations of Differential Geometry
Foundations_of_Differential_Geometry
Class of partial differential equations
In mathematical physics and differential geometry, the Euler–Arnold equations are a class of partial differential equations (PDEs) that describe the geodesic
Euler–Arnold_equation
Description of how spaces intersect in mathematics
to the rank of the vector bundle, and its homology class will be Poincaré dual to the Euler class of the bundle. An extremely special case of this is
Transversality
Spectral sequence in algebraic topology
defined by cupping with the Euler class e ( E ) {\displaystyle e({\mathcal {E}})} . In this case it is given by the top chern class of E {\displaystyle {\mathcal
Serre_spectral_sequence
Method for load calculation in construction
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which
Euler–Bernoulli_beam_theory
Square root of the determinant of a skew-symmetric square matrix
important in the theory of characteristic classes. In particular, it can be used to define the Euler class of a Riemannian manifold that is used in the
Pfaffian
Formula concerning prime numbers
criterion dates from a 1748 paper by Leonhard Euler. The proof uses the fact that the residue classes modulo a prime number are a field. See the article
Euler's_criterion
Quasilinear first-order ordinary differential equation
In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a
Euler's equations (rigid body dynamics)
Euler's_equations_(rigid_body_dynamics)
Short exact sequence of sheaves on projective space
In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of
Euler_sequence
Topological space associated to a vector bundle
Poincaré duality, Euler class of Sphere bundles, Thom classes and Thom isomorphism, and more. Milnor, John. Characteristic classes. is another standard
Thom_space
Movement with a fixed point is rotation
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the body remains
Euler's_rotation_theorem
Analytic function in mathematics
The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable
Riemann_zeta_function
3d hypersurface of degree 5
c({\text{Sym}}^{5}(T^{*}))=\prod _{i=0}^{5}(1+(5-i)\alpha +i\beta )} Then, the Euler class, or the top class is 5 α ( 4 α + β ) ( 3 α + 2 β ) ( 2 α + 3 β ) ( α + 4 β ) 5
Quintic_threefold
Group of languages
68–76. Euler (2022), pp. 25–26. Seebold (1998), p. 13. Euler (2022), pp. 238, 243. Euler (2022), p. 243. Robinson (1992). Euler (2013), p. 53. Euler (2022)
West_Germanic_languages
Concept in contact topology
with the surface Σ {\displaystyle \Sigma } . Moreover, whenever the Euler class e ( ξ ) {\displaystyle e(\xi )} of the contact structure vanishes, the
Rotation_number_(knot_theory)
Extension of the factorial function
}t^{z-1}e^{-t}\,dt} converges absolutely, and is known as the Euler integral of the second kind. (Euler's integral of the first kind is the beta function.) The
Gamma_function
Problem in algebraic geometry
the quotient of N by the normal bundle of i'. Let e(F) be the Euler class (top Chern class) of F, which we view as a homomorphism from Ak−d' (X″) to Ak−d(X″)
Residual_intersection
Probabilistic primality test
also an Euler witness. So each Euler liar gives an Euler witness and so the number of Euler witnesses is larger or equal to the number of Euler liars.
Solovay–Strassen primality test
Solovay–Strassen_primality_test
Brumaire-class submarine
Euler was one of 16 Brumaire-class submarines built for the French Navy during the 1910s. The submarine was assigned to the 2nd Submarine Squadron based
French_submarine_Euler
polynomials, in particular the Vandermonde polynomial. Symmetric polynomial Euler class Giambruno & Zaicev (2005), p. 12. Rather, it only rearranges the other
Alternating_polynomial
Branch of geometry
\mathbb {S} ^{1}} -bundle π : Y → P {\textstyle \pi :Y\rightarrow P} with Euler class [ ω ] / 2 π {\textstyle [\omega ]/2\pi } . Any connection 1-form α {\textstyle
Contact_geometry
On the Euler characteristic of a holomorphic vector bundle on a compact complex manifold
integrated over X is the degree of D, and c1(T(X)) integrated over X is the Euler class 2 − 2g of the curve X, where g is the genus. So we get the classical
Hirzebruch–Riemann–Roch theorem
Hirzebruch–Riemann–Roch_theorem
Infinite products of functions indexed by primes
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product
Euler_product
it is transverse, then we can get a homology cycle by looking at the Euler class of the cokernel bundle E / E ′ {\displaystyle E/E'} over X {\displaystyle
Virtual_fundamental_class
{\displaystyle \mathbb {Q} } of rational numbers is generated by Pontrjagin classes and Euler class: H ∗ ( BSO ( 2 n ) ; Q ) ≅ Q [ p 1 , … , p n , e ] / ( p n −
Classifying_space_for_SO(n)
Type of prime number
prime Euler irregular prime Bernoulli and Euler irregular primes. Factorization of Bernoulli and Euler numbers Factorization of Bernoulli and Euler numbers
Regular_prime
therefore, Chern–Weil theory cannot be used in general to write down the Euler class in terms of the curvature. The conjecture is known to hold in several
Chern's conjecture (affine geometry)
Chern's_conjecture_(affine_geometry)
Right inverse of a fiber bundle map
bundle always has a global section, namely the zero section. However, the Euler class obstructs the existence of a nowhere vanishing section. Obstructions
Section_(fiber_bundle)
Equation used in demography
population growth, probably one of the most important equations is the Euler–Lotka equation. Based on the age demographic of females in the population
Euler–Lotka_equation
Alternative mathematical ordering
Danny (13 December 2004), "Circular groups, planar groups, and the Euler class" (PDF), Geometry & Topology Monographs, 7: 431–491, arXiv:math/0403311
Cyclic_order
Concept in string theory
obstruction bundle, and then realizing the GW invariant as the integral of the Euler class of the obstruction bundle. Making this idea precise requires significant
Gromov–Witten_invariant
Mathematical conjecture
{\displaystyle H^{i}(\mathbb {P} ^{4})} . Using the Euler characteristic and the Euler class, which is the top Chern class, the dimension of this group is 204 {\displaystyle
Mirror_symmetry_conjecture
Principal fiber bundle
H^{2}(M)} . This isomorphism is realized by the Euler class; equivalently, it is the first Chern class of a smooth complex line bundle (essentially because
Circle_bundle
Annual mathematics book award
The Euler Book Prize is an award named after Swiss mathematician and physicist Leonhard Euler (1707–1783) and given annually at the Joint Mathematics
Euler_Book_Prize
Austrian mathematician and philosopher of science (1930–2024)
OCLC 1480369. Clifton, Yeaton H; Smith, J. Wolfgang (November 15, 1963). "The Euler Class as an Obstruction in the Theory of Foliations". Proceedings of the National
Wolfgang_Smith
Species of bird
Euler's flycatcher (Lathrotriccus euleri) is a small passerine bird in the family Tyrannidae, the tyrant flycatchers. It is found on Trinidad and in every
Euler's_flycatcher
Exact homotopy case
{\displaystyle \mathbb {Z} \lbrack c_{1}\rbrack /c_{1}^{k+1}} , where c1 is the Euler class of the U(1)-bundle S2k+1 → CPk, and that the injections CPk → CPk+1,
Classifying_space_for_U(n)
Computation modulo a fixed integer
residue system modulo 4 must have exactly 4 incongruent residue classes. Given the Euler's totient function φ(m), any set of φ(m) integers that are relatively
Modular_arithmetic
Convex polyhedron made from hexagons and pentagons
polyhedron is a dual polyhedron of a geodesic polyhedron. A consequence of Euler's polyhedron formula is that a Goldberg polyhedron always has exactly 12
Goldberg_polyhedron
Way to join two given mathematical manifolds together
isomorphism ψ {\displaystyle \psi } of normal bundles exists whenever their Euler classes are opposite: e ( N M 1 V ) = − e ( N M 2 V ) . {\displaystyle
Connected_sum
Complexity class
give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected, but unproven, that P≠NP, it is unlikely that any
NP-hardness
"only if" part of the Markus conjecture. Chern conjecture (1955) The Euler class of an affine manifold vanishes. Bishop & Goldberg 1968, pp. 223–224.
Affine_manifold
Number divisible only by 1 and itself
the sum of two primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be
Prime_number
Conjecture on zeros of the zeta function
{1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots } Leonhard Euler considered this series in the 1730s for real values of s {\displaystyle
Riemann_hypothesis
20th-century German politician
Ignaz Euler (20 February 1804 – 27 October 1886) was a Prussian notary and politician of the democratic movement [de]. Born in Düsseldorf, Euler was a
Joseph_Euler
Methods used to find numerical solutions of ordinary differential equations
Euler method (or forward Euler method, in contrast with the backward Euler method, to be described below). The method is named after Leonhard Euler who
Numerical methods for ordinary differential equations
Numerical_methods_for_ordinary_differential_equations
The Euler D.I was a German single-seat fighter based on the French Nieuport 11. After seeing the success of the French Nieuport 11 at the front, German
Euler_D.I
Method of integration for rational functions
Euler substitution is a method for evaluating integrals of the form ∫ R ( x , a x 2 + b x + c ) d x , {\displaystyle \int R(x,{\sqrt {ax^{2}+bx+c}})\
Euler_substitution
Every graph has evenly many odd vertices
the number of edges in the graph. Both results were proven by Leonhard Euler (1736) in his famous paper on the Seven Bridges of Königsberg that began
Handshaking_lemma
Mathematical concept in prime numbers
In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible
Idoneal_number
Course designed to prepare students for calculus
particularly in modification and transformation of such expressions. Leonhard Euler wrote the first precalculus book in 1748 called Introductio in analysin
Precalculus
Complexity class used to classify decision problems
complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for
NP_(complexity)
Square array with symbols that each occur once per row and column
The name "Latin square" was inspired by mathematical papers by Leonhard Euler (1707–1783), who used Latin characters as symbols, but any set of symbols
Latin_square
Characteristic class in algebraic topology
H^{2}({\mathbb {C} }P^{n})} be the fundamental class of the hyperplane section. From multiplicativity and the Euler exact sequence for the tangent bundle of
Todd_class
International scientific organization
conferences, publishes a bulletin and awards a number of medals, including the Euler, Hall, Kirkman, and Stanton Medals. It is based in Duluth, Minnesota and
Institute of Combinatorics and its Applications
Institute_of_Combinatorics_and_its_Applications
Branch of mathematics
17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably
Topology
Odd composite number which passes the given congruence
In mathematics, an odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and a ( n − 1 ) / 2 ≡ ± 1 ( mod n ) {\displaystyle
Euler_pseudoprime
Number equal to the sum of its proper divisors
Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form. This is known as the Euclid–Euler theorem. It is not known whether
Perfect_number
211, 2311, 200560490131 (OEIS: A018239) Euler irregular primes are primes p {\displaystyle p} that divide an Euler number E 2 n , {\displaystyle E_{2n},}
List_of_prime_numbers
Country in South Asia
most certainly I have never met his equal, and I can compare him only with Euler and Jacobi. He worked, far more than the majority of modern mathematicians
India
theory Herbrand–Ribet theorem Vandiver's conjecture Stickelberger's theorem Euler system p-adic L-function Arithmetic geometry Complex multiplication Abelian
List of algebraic number theory topics
List_of_algebraic_number_theory_topics
Meromorphic function on the complex plane
Fundamental subclasses of L-functions were built on the work of Leonhard Euler (which is now known as the Riemann zeta function). Most notably, the mathematicians
L-function
Used to count, measure, and label
would later be named Euler's number (e). Irrational numbers began to be studied systematically in the 18th century, with Leonhard Euler who proved that the
Number
Seven mathematical problems with a US$1 million prize for each solution
projective variety. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X. The
Millennium_Prize_Problems
Mathematical concept
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. When k
Lucky_numbers_of_Euler
Number, approximately 3.14
"Estimating π" (PDF). How Euler Did It. Reprinted in How Euler Did Even More. Mathematical Association of America. 2014. pp. 109–118. Euler, Leonhard (1755).
Pi
American radio and television personality (born 1954)
original on October 8, 2016. Retrieved August 17, 2016. Cassidy, Grace; Euler, Laura (May 23, 2018). "Hamptons Celebrity Homes Mapped". Curbed. Archived
Howard_Stern
Mathematical problem
orthogonal Latin squares is Graeco-Latin square, introduced by Euler. A Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order n over
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
American annual mathematics conference
Shing-Tung Yau, Some theorems in Kähler geometry Jeff Cheeger, Transgressed Euler classes of SL(2n,Z)-bundles and adiabatic limits of eta-invariants Chris Croke
Geometry_Festival
Unsolved problem in computer science
input to the algorithm. The general class of questions that some algorithm can answer in polynomial time is "P" or "class P". For some questions, there is
P_versus_NP_problem
Mathematical theorem
holomorphic Euler characteristic, c12 = (K.K) is a Chern number and the self-intersection number of the canonical class K, and e = c2 is the topological Euler characteristic
Riemann–Roch theorem for surfaces
Riemann–Roch_theorem_for_surfaces
Condition involving social and behavioral differences
are prescribed psychoactive drugs or anticonvulsants. Commonly used drug classes include antidepressants, stimulants, and antipsychotics. Among antipsychotics
Autism
Country primarily in Western Europe
Country profile: France Archived 1 October 2020 at the Wayback Machine, Euler Hermes "These are the top 10 manufacturing countries in the world". World
France
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