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 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 a second-order polynomial,, all the roots are in the left half plane ( and the system with characteristic equation is stable ) if all the coefficients satisfy.

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 a vector with linear addressing, the element with index i is located at the address B + c · i, where B is a fixed base address and c a fixed constant, sometimes called the address increment or stride.

For the case of an object that is small compared with the radial distance to its axis of rotation, such as a tin can swinging from a long string or a planet orbiting in a circle around the Sun, the angular momentum can be expressed as its linear momentum,, crossed by its position from the origin, r. Thus, the angular momentum L of a particle with respect to some point of origin is

For example, in the Schrödinger picture, there is a linear operator U with the property that if an electron is in state right now, then in one minute it will be in the state, the same U for every possible.

For example, observable physical quantities are represented by self-adjoint operators, such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotation or the progression of time.

For example, the condition number associated with the linear equation

For simplicity, the following descriptions focus on continuous-time and discrete-time linear systems.

For any n, the divergence is a linear operator, and it satisfies the " product rule "

For the statistics usage, see hierarchical linear modeling.

For a flow, the vector field Φ ( x ) is a linear function of the position in the phase space, that is,

For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category.

For other ways to solve this kind of equations, see below, System of linear equations.

For example, according to ancient Hebrew belief, life takes a linear ( and not cyclical ) path ; the world began with God and is constantly headed toward God ’ s final goal for creation.

For an overview of a linear implementation of this framework, see linear regression.

For other examples of quotient objects, see quotient ring, quotient space ( linear algebra ), quotient space ( topology ), and quotient set.

For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

For molecules with N atoms in them, linear molecules have 3N – 5 degrees of vibrational modes, whereas nonlinear molecules have 3N – 6 degrees of vibrational modes ( also called vibrational degrees of freedom ).

For every finite dimensional matrix Lie algebra, there is a linear group ( matrix Lie group ) with this algebra as its Lie algebra.

For a population of size n, with a sequence of values y < sub > i </ sub >, i = 1 to n, that are indexed in non-decreasing order ( y < sub > i </ sub > ≤ y < sub > i + 1 </ sub >), the Lorenz curve is the continuous piecewise linear function connecting the points ( F < sub > i </ sub >, L < sub > i </ sub > ), i

For linear relations, regression analyses here are based on forms of the general linear model.

For example, naive Bayes and linear discriminant analysis are joint probability models, whereas logistic regression is a conditional probability model.

For example, consider the case where the function is a linear function of the form

For example, bacterial chromosomes, plasmids, mitochondrial DNA and chloroplast DNA are usually circular double-stranded DNA molecules, while chromosomes of the eukaryotic nucleus are usually linear double-stranded DNA molecules.

For example, in classical mechanics, the derivative is used ubiquitously, and in quantum mechanics, observables are represented by linear operators.