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If M^3 is a closed oriented connected 3-manifold such that every simple closed curve in M lies interior to a ball in M, then M is homeomorphic with the hypersphere, S^3.

A relation between compact boundaryless manifolds (also called closed manifolds). Two closed manifolds are bordant iff their disjoint union is the boundary of a compact ...

While some authors define "circumference" as distance around an arbitrary closed object (sometimes restricted to a closed curved object), in the work, the term "perimeter" is ...

A subset of a topological space is called clopen if it is both closed and open.

The quartic surface resembling a squashed round cushion on a barroom stool and given by the equation z^2x^2-z^4-2zx^2+2z^3+x^2-z^2 -(x^2-z)^2-y^4-2x^2y^2-y^2z^2+2y^2z+y^2=0.

The double sphere is the degenerate quartic surface (x^2+y^2+z^2-r^2)^2=0 obtained by squaring the left-hand side of the equation of a usual sphere x^2+y^2+z^2-r^2=0.

A closed ideal I in a C^*-algebra A is called essential if I has nonzero intersection with every other nonzero closed ideal A or, equivalently, if aI={0} implies a=0 for all ...

As used in physics, the term "exact" generally refers to a solution that captures the entire physics and mathematics of a problem as opposed to one that is approximate, ...

Let A=a_(ij) be a matrix with positive coefficients so that a_(ij)>0 for all i,j=1, 2, ..., n, then A has a positive eigenvalue lambda_0, and all its eigenvalues lie on the ...

Consider two closed oriented space curves f_1:C_1->R^3 and f_2:C_2->R^3, where C_1 and C_2 are distinct circles, f_1 and f_2 are differentiable C^1 functions, and f_1(C_1) ...