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A symplectic form on a smooth manifold M is a smooth closed 2-form omega on M which is nondegenerate such that at every point m, the alternating bilinear form omega_m on the ...

The study of angles and of the angular relationships of planar and three-dimensional figures is known as trigonometry. The trigonometric functions (also called the circular ...

An expression in mathematics which does not have meaning and so which is not assigned an interpretation. For example, division by zero is undefined in the field of real ...

The series sum_(j=1)^(infty)f_j(z) is said to be uniformly Cauchy on compact sets if, for each compact K subset= U and each epsilon>0, there exists an N>0 such that for all ...

A normed vector space X=(X,||·||_X) is said to be uniformly convex if for sequences {x_n}={x_n}_(n=1)^infty, {y_n}={y_n}_(n=1)^infty, the assumptions ||x_n||_X<=1, ...

An unknown is an independent variable of a function. A function with n independent variables is said to be a function "in," "with," or "of" n unknowns.

A function of one or more variables whose range is three-dimensional (or, in general, n-dimensional), as compared to a scalar function, whose range is one-dimensional. Vector ...

The W-transform of a function f(x) is defined by the integral where Gamma[(beta_m)+s, 1-(alpha_n)-s; (alpha_p^(n+1))+s, 1-(beta_q^(m+1))-s] =Gamma[beta_1+s, ..., beta_m+s, ...

Maps between CW-complexes that induce isomorphisms on all homotopy groups are actually homotopy equivalences.

If f is a function on an open set U, then the zero set of f is the set Z={z in U:f(z)=0}. A subset of a topological space X is called a zero set if it is equal to f^(-1)(0) ...