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The number of real roots of an algebraic equation with real coefficients whose real roots are simple over an interval, the endpoints of which are not roots, is equal to the ...

Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time. The term ...

Smale's problems are a list of 18 challenging problems for the twenty-first century proposed by Field medalist Steven Smale. These problems were inspired in part by Hilbert's ...

The Kuhn-Tucker theorem is a theorem in nonlinear programming which states that if a regularity condition holds and f and the functions h_j are convex, then a solution ...

Taylor's theorem states that any function satisfying certain conditions may be represented by a Taylor series, Taylor's theorem (without the remainder term) was devised by ...

The term "parameter" is used in a number of ways in mathematics. In general, mathematical functions may have a number of arguments. Arguments that are typically varied when ...

The algebraic identity (sum_(i=1)^na_ic_i)(sum_(i=1)^nb_id_i)-(sum_(i=1)^na_id_i)(sum_(i=1)^nb_ic_i) =sum_(1<=i<j<=n)(a_ib_j-a_jb_i)(c_id_j-c_jd_i). (1) Letting c_i=a_i and ...

The Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques. To integrate a function f(x) over some interval [a,b], ...

Relationships between the number of singularities of plane algebraic curves. Given a plane curve, m = n(n-1)-2delta-3kappa (1) n = m(m-1)-2tau-3iota (2) iota = ...

A special case of Hölder's sum inequality with p=q=2, (sum_(k=1)^na_kb_k)^2<=(sum_(k=1)^na_k^2)(sum_(k=1)^nb_k^2), (1) where equality holds for a_k=cb_k. The inequality is ...