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 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 example, it is evinced by the adolescent ( or adult ) whose beliefs and actions represent primarily his rebellion and reaction against the ideas and behavior patterns of others, rather than his inner conviction and choice.

For some this marks the beginning of the " mature " Classical style, where the period of reaction against the complexity of the late Baroque began to be replaced with a period of integration of elements of both Baroque and Classical styles.

For any reaction mixture to exist at equilibrium, the rates of the forward and backward ( reverse ) reactions are equal.

For example, adding more S from the outside will cause an excess of products, and the system will try to counteract this by increasing the reverse reaction and pushing the equilibrium point backward ( though the equilibrium constant will stay the same ).

For example, if in a search task the reaction times vary proportionally with the number of elements, then it is evident that this cognitive process of searching involves serial instead of parallel processing.

For example, an increase in pressure due to decreasing volume causes the reaction to shift to the side with the fewer moles of gas.

* For some reactions, the presence of electromagnetic radiation, most notably ultraviolet light, is needed to promote the breaking of bonds to start the reaction.

For example, the thermite reaction is used to generate light and heat in pyrotechnics and welding.

For example, the probability that an alpha particle striking a beryllium target will produce a neutron can be expressed as the equivalent cross section of beryllium for this type of reaction.

For example, depending on surface porosity, 5 – 30 atomic layers of titanium carbide are hydrolyzed within 5 minutes at ambient conditions, following by saturation of the reaction.

For a general reaction

For example, on reaction between potassium permanganate and sodium sulfite.

For example, the sentence " The coat is red " has no observer, the sentence " We see the coat as red " ( where " we " indicates observers ) appears more specific in context as regards light waves and colour as determined by modern science, that is, colour results from a reaction in the human brain.

For an exothermic reaction at constant pressure, the system's change in enthalpy equals the energy released in the reaction, including the energy retained in the system and lost through expansion against its surroundings.

For Catalan historian Xavier Diez, individualist anarchism " under its iconoclastic, antiintelectual, antitheist run, which goes against all sacralized ideas or values it entailed, a philosophy of life which could be considered a reaction against the sacred gods of capitalist society.

For the five 6. 4 meganewton sea-level thrust F-1 rocket engines of the Saturn V, burning together, the reaction generated roughly 1. 62 × 10 < sup > 11 </ sup > watts ( J / s ) ( 162 gigawatt ) or 217 million horsepower.

For Greenberg, modernism thus formed a reaction against the development of such examples of modern consumer culture as commercial popular music, Hollywood, and advertising.

For example, an alchemical paradigm might describe the act of wood burning as the wood " releasing its essence of elemental Fire ," while modern science would describe fire as " combustion resulting from a complex chemical reaction.

For example, an explosion is the result of a chain reaction that transforms potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation.

For certain of the heaviest nuclei which produce neutrons on fission, and which also easily absorb neutrons to initiate fission, a self-igniting type of neutron-initiated fission can be obtained, in a so-called chain reaction.

For this reason, in early experiments a special reaction channel was chosen to facilitate the identification: the interaction of an antineutrino with one of the hydrogen nuclei in the water molecules.

For example, the Chernobyl disaster involved a runaway chain reaction but the result was a low-powered steam explosion from the relatively small release of heat, as compared with a bomb.

For example, power plants licensed in the United States require a negative void coefficient of reactivity ( this means that if water is removed from the reactor core, the nuclear reaction will tend to shut down, not increase ).

For example, a carbonyl compound can be used as a nucleophile by converting it into an enolate, or as an electrophile ; the combination of the two is called the aldol reaction.