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If P(x) is an irreducible cubic polynomial all of whose roots are real, then to obtain them by radicals, you must take roots of nonreal numbers at some point.

Let X and Y be CW-complexes and let X_n (respectively Y_n) denote the n-skeleton of X (respectively Y). Then a continuous map f:X->Y is said to be cellular if it takes ...

Characteristic classes are cohomology classes in the base space of a vector bundle, defined through obstruction theory, which are (perhaps partial) obstructions to the ...

Chrystal's identity is the algebraic identity ((b-c)^2+(b+c)^2+2(b^2-c^2))/((b^4-2b^2c^2+c^4)[1/((b-c)^2)+2/(b^2-c^2)+1/((b+c)^2)])=1 given as an exercise by Chrystal (1886).

The surface given by the parametric equations x = e^(bv)cosv+e^(av)cosucosv (1) y = e^(bv)sinv+e^(av)cosusinv (2) z = e^(av)sinu. (3) For a=b=1, the coefficients of the first ...

A cubic equation is an equation involving a cubic polynomial, i.e., one of the form a_3x^3+a_2x^2+a_1x+a_0=0. Since a_3!=0 (or else the polynomial would be quadratic and not ...

A connection defined on a smooth algebraic variety defined over the complex numbers.

Let X be a continuum (i.e., a compact connected metric space). Then X is hereditarily unicoherent provided that every subcontinuum of X is unicoherent. Any hereditarily ...

The homeomorphism group of a topological space X is the set of all homeomorphisms f:X->X, which forms a group by composition.

Let C denote a chain complex, a portion of which is shown below: ...->C_(n+1)->C_n->C_(n-1)->.... Let H_n(C)=kerpartial_n/Impartial_(n+1) denotes the nth homology group. Then ...