topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that...

PDEs. Otherwise, Euler's equation may refer to a non-differential equation, as in these three cases: Euler–Lotka equation, a characteristic equation employed...

geometry, the Euler characteristic of an orbifold, or orbifold Euler characteristic, is a generalization of the topological Euler characteristic that includes...

fibration becomes trivial after taking a finite cover of B. The orbifold Euler characteristic χ ( B ) {\displaystyle \chi (B)} of the orbifold B is given by χ...

polyhedron equals 2, a number now commonly known as the Euler characteristic. In the field of physics, Euler reformulated Newton's laws of physics into new laws...

eigenvector of a matrix Characteristic word, a subclass of Sturmian word Euler characteristic, a topological invariant Method of characteristics, a technique for...

Galois cohomology, the local Euler characteristic formula is a result due to John Tate that computes the Euler characteristic of the group cohomology of...

important examples of characteristic numbers are Stiefel–Whitney numbers, Chern numbers, Pontryagin numbers, and the Euler characteristic. Given an oriented...

cohomology and singular cohomology such as Hodge theory, and formulas on Euler characteristics in coherent sheaf cohomology such as the Riemann–Roch theorem. In...

is the Euler characteristic of M {\displaystyle M} . A particularly useful corollary is when there is a non-vanishing vector field implying Euler characteristic...

E = 2 edges, and F = 1 face. Thus the Euler characteristic of the torus is 1 − 2 + 1 = 0. The Euler characteristic of other surfaces is a useful topological...

ds is the line element along the boundary of M. Here, χ(M) is the Euler characteristic of M. If the boundary ∂M is piecewise smooth, then we interpret the...

respectively in two and three dimensions. Attempts to generalise the Euler characteristic of polyhedra to higher-dimensional polytopes led to the development...

The Euler characteristic, a topological invariant. The receiver operating characteristic in statistical decision theory. The point characteristic function...

The 18th-century Swiss mathematician Leonhard Euler (1707–1783) is among the most prolific and successful mathematicians in the history of the field....

the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In...

All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even Euler characteristic may or may not be orientable. For...

reason that the Euler characteristic has a definition in terms of homology groups; see below for the relation to the Euler characteristic). In the particular...

also contain tetrahedral symmetry. The five Platonic solids have an Euler characteristic of 2. This simply reflects that the surface is a topological 2-sphere...

all of the indices at all of the zeros must be two, because the Euler characteristic of the 2-sphere is two. Therefore, there must be at least one zero...

[citation needed] For any polyhedron homeomorphic to a sphere, the Euler characteristic is χ = V − E + F = 2 {\displaystyle \chi =V-E+F=2} , where V {\displaystyle...

Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler–Poincaré characteristic (a topological invariant defined as the alternating sum of the...

}(-1)^{i}b_{i}(K,F),\,} where χ ( K ) {\displaystyle \chi (K)} denotes Euler characteristic of K and any field F. For any two spaces X and Y we have P X × Y...

family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of k of...

in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted"...

Refutations, Cambridge University Press (1976) - discussion of proof of Euler characteristic Wenninger, Magnus (1983). Dual Models. Cambridge University Press...

electrophoresis. In neuroscience, topological quantities like the Euler characteristic and Betti number have been used to measure the complexity of patterns...

the Euler characteristic of the total space is given by: χ ( E ) = χ ( B ) χ ( F ) . {\displaystyle \chi (E)=\chi (B)\chi (F).} Here the Euler characteristics...

{\displaystyle \int _{\Sigma }K\,dA=2\pi \chi (\Sigma )} where χ(Σ) is the Euler characteristic, which is an integer. An example is the surface area of a sphere...

holes). So in this case, the Euler characteristic is -1. To bring this into the discrete world, the Euler characteristic of a mesh is computed in terms...