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A topological space X is pathwise-connected iff for every two points x,y in X, there is a continuous function f from [0,1] to X such that f(0)=x and f(1)=y. Roughly speaking, ...

The alternative hypothesis is the hypothesis used in hypothesis testing that is contrary to the null hypothesis. It is usually taken to be that the observations are the ...

A set is said to be bounded from above if it has an upper bound. Consider the real numbers with their usual order. Then for any set M subset= R, the supremum supM exists (in ...

A set is said to be bounded from below if it has a lower bound. Consider the real numbers with their usual order. Then for any set M subset= R, the infimum infM exists (in R) ...

The definite integral int_a^bx^ndx={(b^(n+1)-a^(n+1))/(n+1) for n!=1; ln(b/a) for n=-1, (1) where a, b, and x are real numbers and lnx is the natural logarithm.

A complex rotation is a map of the form z|->ze^(itheta), where theta is a real number, which corresponds to counterclockwise rotation by theta radians about the origin of ...

Let h be a real-valued harmonic function on a bounded domain Omega, then the Dirichlet energy is defined as int_Omega|del h|^2dx, where del is the gradient.

An absolutely continuous measure on partialD whose density has the form exp(x+y^_), where x and y are real-valued functions in L^infty, ||y||_infty<pi/2, exp is the ...

An infinitesimal is some quantity that is explicitly nonzero and yet smaller in absolute value than any real quantity. The understanding of infinitesimals was a major ...

A spectrum formed by the Lagrange numbers. The only ones less than three are the Lagrange numbers, but the last gaps end at Freiman's constant. Real numbers larger than ...