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For what value of x is f(x)=x^(1/x) a maximum? The maximum occurs at x=e, where f^'(x)=x^(-2+1/x)(1-lnx)=0, (1) which is zero at x=e and gives a maximum of ...

A tetraview is a visualization technique for bivariate complex functions. In the simplest case, the graph of a complex-valued function w=w(z) can be considered as a ...

Every closed three-manifold with finite fundamental group has a metric of constant positive scalar curvature, and hence is homeomorphic to a quotient S^3/Gamma, where Gamma ...

The tribox, also called the Penrose rectangle or Penrose square, is an impossible figure that is the generalization of the Penrose triangle from a triangle to a square. ...

int_0^(pi/2)cos^nxdx = int_0^(pi/2)sin^nxdx (1) = (sqrt(pi)Gamma(1/2(n+1)))/(nGamma(1/2n)) (2) = ((n-1)!!)/(n!!){1/2pi for n=2, 4, ...; 1 for n=3, 5, ..., (3) where Gamma(n) ...

Let N be an odd integer, and assume there exists a Lucas sequence {U_n} with associated Sylvester cyclotomic numbers {Q_n} such that there is an n>sqrt(N) (with n and N ...

If an n-sided Poncelet transverse constructed for two given conic sections is closed for one point of origin, it is closed for any position of the point of origin. ...

The second Mersenne prime M_3=2^3-1, which is itself the exponent of Mersenne prime M_7=2^7-1=127. It gives rise to the perfect number P_7=M_7·2^6=8128. It is a Gaussian ...

Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these ...

The degree to which a given quantity is correct and free from error. For example, a quantity specified as 100+/-1 has an (absolute) accuracy of +/-1 (meaning its true value ...