For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, could be obtained by means of Newton's second law and Hooke's law.

For instance, retaining the first two terms of the series yields the second-order approximation to f ’( x ) mentioned at the end of the section Higher-order differences.

For example, the sentence involving Napoleon can be rewritten as “ any group of people that includes me and the parents of each person in the group must also include Napoleon ,” which is easily interpreted as a statement in second-order logic ( one would naturally start by assigning a name, such as G, to the group of people under consideration ).

For example, a second-order Butterworth filter will reduce the signal amplitude to one fourth its original level every time the frequency doubles ( so power decreases by 12 dB per octave, or 40 dB per decade ).

For second-order divisions ( under provinces and special administrative regions ), there are counties, provincial cities ( 56 ), bureaus ( 34 ) and management bureaus ( 7 ).

For an infinitesimal deformation the displacements and the displacement gradients are small compared to unity, i. e., and, allowing for the geometric linearisation of the Lagrangian finite strain tensor, and the Eulerian finite strain tensor, i. e. the non-linear or second-order terms of the finite strain tensor can be neglected.

For example, money satisfies no biological or psychological needs, but a pay check appears to reduce drive through second-order conditioning.

For example, while the analysis of a table ( matrix, or second-order arry ) of data is routine in several fields, multiway methods are applied to data sets that involve 3rd, 4th, or higher-orders.

For example, second-order arithmetic can express the principle " Every countable vector space has a basis " but it cannot express the principle " Every vector space has a basis ".

For example, the second-order sentence says that for every set P of individuals and every individual x, either x is in P or it is not ( this is the principle of bivalence ).

For example, if the domain is the set of all real numbers, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀ x ∃ y ( x + y = 0 ) but one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum.

For example, the second-order derivative would be:

For example, a second-order Butterworth filter, which has maximally flat passband frequency response, has a of.

For instance, a second-order low-pass notch filter section only reduces ( rather than eliminates ) very high frequencies, but has a steep response falling to zero at a specific frequency ( the so-called notch frequency ).

For a boundary value problem of a second-order ordinary differential equation, the method is stated as follows.

For example, the Löwenheim number of second-order logic is already larger than the first measurable cardinal, if such a cardinal exists.

" It is possible to define a formula True ( n ) whose extension is T *, but only by drawing on a metalanguage whose expressive power goes beyond that of L. For example, a truth predicate for first-order arithmetic can be defined in second-order arithmetic.

For a second-order linear autonomous systems, a critical point is a saddle point if the characteristic equation has one positive and one negative real eigenvalue.

For comparison, in the equivalent Euler – Lagrange equations of motion of Lagrangian mechanics, the conjugate momenta also do not appear ; however, those equations are a system of N, generally second-order equations for the time evolution of the generalized coordinates.

For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic.

For a second-order reaction half-lives progressively double.

For example, an animal might first learn to associate a bell with food ( first-order conditioning ), but then learn to associate a light with the bell ( second-order conditioning ).

For example, if MK is consistent then it has a countable first-order model, while second-order ZFC has no countable models.

For example, consider the quadratic polynomial

For example, one can use it to determine, for any polynomial equation, whether it has a solution by radicals.

* K, the ring of polynomials over a field K. For each nonzero polynomial P, define f ( P ) to be the degree of P.

For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time.

For example, P < sup > SAT </ sup > is the class of problems solvable in polynomial time by a deterministic Turing machine with an oracle for the Boolean satisfiability problem.

For example, is a polynomial, but is not, because its second term involves division by the variable x ( 4 / x ), and also because its third term contains an exponent that is not a non-negative integer ( 3 / 2 ).

For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences ; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science ; they are used in calculus and numerical analysis to approximate other functions.

For example, the following is a polynomial:

For example, ( x + 1 )< sup > 3 </ sup > is a polynomial ; its standard form is x < sup > 3 </ sup > + 3x < sup > 2 </ sup > + 3x + 1.

For example, x < sup > 3 </ sup >/ 12 is considered a valid term in a polynomial ( and a polynomial by itself ) because it is equivalent to 1 / 12x < sup > 3 </ sup > and 1 / 12 is just a constant.

For polynomials in more than one variable the notion of root does not exist, and there are usually infinitely many combinations of values for the variables for which the polynomial function takes the value zero.

For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions.

For the polynomial function

For instance, the ring ( in fact field ) of complex numbers, which can be constructed from the polynomial ring R over the real numbers by factoring out the ideal of multiples of the polynomial.

For f a real polynomial in x, and for any a in such an algebra define f ( a ) to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that ( fg ) ( a )

For some problems, quantum computers offer a polynomial speedup.

The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L < sup > 2 </ sup >, where L is the number of bits in the number to be factored ; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 10 < sup > 4 </ sup > qubits without error correction.

For every, there exists a polynomial function p over C such that for all x in, we have, or equivalently, the supremum norm.

For example, the square root of 2 is irrational and not transcendental ( because it is a solution of the polynomial equation x < sup > 2 </ sup > − 2 = 0 ).

( For example if the variables x, y, and z are permuted in all 6 possible ways in the polynomial x + y-z then we get a total of 3 different polynomials: x + y − z, x + z-y, and y + z − x.

For instance, the problem of integer factorization is suspected to require more than a polynomial function of the length of the input as run-time if the input is given in binary, but it only needs linear runtime if the input is presented in unary.

For example, to compute one prime factor of the natural number N in polynomial time ( no polynomial time factorization algorithm is known in traditional complexity theory ; see integer factorization ):