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Let Jones and Smith be the only two contestants in an election that will end in a deadlock when all votes for Jones (J) and Smith (S) are counted. What is the expectation ...

Generally, a face is a component polygon, polyhedron, or polytope. A two-dimensional face thus has vertices and edges, and can be used to make cells. More formally, a face is ...

If P(x,y) and P(x^',y^') are two points on an ellipse (x^2)/(a^2)+(y^2)/(b^2)=1, (1) with eccentric angles phi and phi^' such that tanphitanphi^'=b/a (2) and A=P(a,0) and ...

Let f:R×R->R be a one-parameter family of C^3 maps satisfying f(0,0) = 0 (1) [(partialf)/(partialx)]_(mu=0,x=0) = -1 (2) [(partial^2f)/(partialx^2)]_(mu=0,x=0) < 0 (3) ...

A finite, increasing sequence of integers {n_1,...,n_m} such that sum_(i=1)^m1/(n_i)-product_(i=1)^m1/(n_i) in N. A sequence is a Giuga sequence iff it satisfies ...

One of the eight convex deltahedra built up from 16 equilateral triangles. It consists of two oppositely faced square pyramids rotated 45 degrees to each other and separated ...

The Kronecker sum is the matrix sum defined by A direct sum B=A tensor I_b+I_a tensor B, (1) where A and B are square matrices of order a and b, respectively, I_n is the ...

Given a function of two variables df = (partialf)/(partialx)dx+(partialf)/(partialy)dy (1) = udx+vdy, (2) change the differentials from dx and dy to du and dy with the ...

A topological space is locally connected at the point x if every neighborhood of x contains a connected open neighborhood. It is called locally connected if it is locally ...

The number 1000000=10^6. While one million in the "American" system of numbers means the same thing as one million in the "British" system, the words billion, trillion, etc., ...