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The general problem was solved in the affirmative, in 1927, by Emil Artin, [6] for positive semidefinite functions over the reals or more generally real-closed fields. An algorithmic solution was found by Charles Delzell in 1984. [ 7 ]
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients [1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.
The question is whether or not, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), an algorithm can also find that solution quickly. Since the former describes the class of problems termed NP, while the latter describes P, the question is equivalent to asking whether all problems in NP are ...
A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant c. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or ...
Given a polynomial equation or a system of polynomial equations it is often useful to compute or to bound the number of solutions without computing explicitly the solutions. In the case of a single equation, this problem is solved by the fundamental theorem of algebra, which asserts that the number of complex solutions is bounded by the degree ...
The solution of the complex differential equation can be found as (), from which the real part is the solution of the original equation. Complex replacement is used for solving differential equations when the non-homogeneous term is expressed in terms of a sinusoidal function or an exponential function, which can be converted into a complex ...
One may now proceed as in the differential equation case, since the general solution of an N-th order linear difference equation is also the linear combination of N linearly independent solutions. Applying reduction of order in case of a multiple root m 1 will yield expressions involving a discrete version of ln , φ ( n ) = ∑ k = 1 n 1 k − ...
Some solutions of a differential equation having a regular singular point with indicial roots = and .. In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form ″ + ′ + = with ′ and ″.