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On a measure space X, the set of square integrable L2-functions is an L^2-space. Taken together with the L2-inner product with respect to a measure mu, <f,g>=int_Xfgdmu (1) ...

A function f:X->R is measurable if, for every real number a, the set {x in X:f(x)>a} is measurable. When X=R with Lebesgue measure, or more generally any Borel measure, then ...

Related to or being the mathematically most simple case. More generally, the word "trivial" is used to describe any result which requires little or no effort to derive or ...

The curve produced by fixed point P on the circumference of a small circle of radius b rolling around the inside of a large circle of radius a>b. A hypocycloid is therefore a ...

A curve of constant width constructed by drawing arcs from each polygon vertex of an equilateral triangle between the other two vertices. The Reuleaux triangle has the ...

The power series that defines the exponential map e^x also defines a map between matrices. In particular, exp(A) = e^(A) (1) = sum_(n=0)^(infty)(A^n)/(n!) (2) = ...

The tensor product of two vector spaces V and W, denoted V tensor W and also called the tensor direct product, is a way of creating a new vector space analogous to ...

Continuum percolation can be thought of as a continuous, uncountable version of percolation theory-a theory which, in its most studied form, takes place on a discrete, ...

For vectors u=(u_x,u_y,u_z) and v=(v_x,v_y,v_z) in R^3, the cross product in is defined by uxv = x^^(u_yv_z-u_zv_y)-y^^(u_xv_z-u_zv_x)+z^^(u_xv_y-u_yv_x) (1) = ...

A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight ...