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if X is a locally compact space, and one uses Euler characteristics with compact supports, no assumptions on M or N are needed. if X is a stratified space all of whose strata are even-dimensional, the inclusion–exclusion principle holds if M and N are unions of strata. This applies in particular if M and N are subvarieties of a complex ...
In differential geometry, the Euler characteristic of an orbifold, ... "On equivariant Euler characteristics". Journal of Geometry and Physics. 6 (4): 671–677.
Thus the Euler class is a generalization of the Euler characteristic to vector bundles other than tangent bundles. In turn, the Euler class is the archetype for other characteristic classes of vector bundles, in that each "top" characteristic class equals the Euler class, as follows. Modding out by 2 induces a map
Leonhard Euler (/ ˈ ɔɪ l ər / OY-lər; [b] German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] ⓘ, Swiss Standard German: [ˈleɔnhard ˈɔʏlər]; 15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of ...
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case.
Euler's Gem: The Polyhedron Formula and the Birth of Topology is a book on the formula + = for the Euler characteristic of convex polyhedra and its connections to the history of topology. It was written by David Richeson and published in 2008 by the Princeton University Press , with a paperback edition in 2012.
Leonhard Euler is credited of introducing both specifications in two publications written in 1755 [3] and 1759. [4] [5] Joseph-Louis Lagrange studied the equations of motion in connection to the principle of least action in 1760, later in a treaty of fluid mechanics in 1781, [6] and thirdly in his book Mécanique analytique. [5]
Generally, the Euler equations are solved by Riemann's method of characteristics. This involves finding curves in plane of independent variables (i.e., x {\displaystyle x} and t {\displaystyle t} ) along which partial differential equations (PDEs) degenerate into ordinary differential equations (ODEs).