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If xsinalpha=sin(2beta-alpha), then (1+x)int_0^alpha(dphi)/(sqrt(1-x^2sin^2phi))=2int_0^beta(dphi)/(sqrt(1-(4x)/((1+x)^2)sin^2phi)).

A two-dimensional map similar to the Hénon map but with the term -alphax_n^2 replaced by -alpha|x_n|. It is given by the equations x_(n+1) = 1-alpha|x_n|+y_n (1) y_(n+1) = ...

Let f(z) be an analytic function in an angular domain W:|argz|<alphapi/2. Suppose there is a constant M such that for each epsilon>0, each finite boundary point has a ...

The system of partial differential equations del ^4u = E(v_(xy)^2-v_(xx)v_(yy)) (1) del ^4v = alpha+beta(u_(yy)v_(xx)+u_(xx)v_(yy)-2u_(xy)v_(xy)), (2) where del ^4 is the ...

There are several types of integrals which go under the name of a "Dirichlet integral." The integral D[u]=int_Omega|del u|^2dV (1) appears in Dirichlet's principle. The ...

The join G=G_1+G_2 of graphs G_1 and G_2 with disjoint point sets V_1 and V_2 and edge sets X_1 and X_2 is the graph union G_1 union G_2 together with all the edges joining ...

Integration by parts is a technique for performing indefinite integration intudv or definite integration int_a^budv by expanding the differential of a product of functions ...

A maximal independent vertex set of a graph is an independent vertex set that cannot be expanded to another independent vertex set by addition of any vertex in the graph. A ...

Let s_i be the sum of the products of distinct polynomial roots r_j of the polynomial equation of degree n a_nx^n+a_(n-1)x^(n-1)+...+a_1x+a_0=0, (1) where the roots are taken ...

Given a commutative unit ring R and a filtration F:... subset= I_2 subset= I_1 subset= I_0=R (1) of ideals of R, the associated graded ring of R with respect to F is the ...