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...

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

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

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...

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

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

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...

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

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

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...

}(-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...

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...

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

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

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...

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

[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...

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

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...

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

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...

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

(not to be confused with the 0 and 1 of the ring of scalars). The Euler characteristic of a bounded finite poset is μ(0,1). The reason for this terminology...

odd-dimensional manifold M has Euler characteristic zero, which in turn gives that any manifold that bounds has even Euler characteristic. Poincaré duality is closely...

handles on it. Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is...