Sometimes slightly stronger theories such as Morse-Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe are used, but in fact most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic.

This theory, which combined Landau's theory of second-order phase transitions with a Schrödinger-like wave equation, had great success in explaining the macroscopic properties of superconductors.

A logical characterization of PSPACE from descriptive complexity theory is that it is the set of problems expressible in second-order logic with the addition of a transitive closure operator.

However, unlike standard Newtonian mechanics, the initial velocity field is already specified by which is a symptom of this being a first-order theory, not a second-order theory.

The proof of this theorem follows from the theory of ordinary differential equations, by noticing that the geodesic equation is a second-order ODE.

The standard exposition of perturbation theory is given in terms of the order to which the perturbation is carried out: first-order perturbation theory or second-order perturbation theory, and whether the perturbed states are degenerate ( that is, singular ), in which case extra care must be taken, and the theory is slightly more difficult.

The process of converting the narrative form of a scientific theory into second-order logic is commonly called " Ramsification " ( sometimes also spelled " Ramseyfication ").

* Discrete structural analysis: whereas neoclassical theory uses continuous marginal modes of analysis in order to achieve second-order economizing ( adjusting margins ), TCE analyzes the basic structures of the firm and its governance in order to achieve first-order economizing ( improving the basic governance structure ).

This was still a second-order axiomatization ( expressing induction in terms of arbitrary subsets, thus with an implicit use of set theory ) as concerns for expressing theories in first-order logic were not yet understood.

Defined in this way, holons are related to the concept of autopoiesis, especially as it was developed in the application of Stafford Beer to second-order cybernetics and viable system theory, but also Niklas Luhmann in his social systems theory.

The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the Kondo problem, in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and critical phenomena in 1971.

In 1969, Rabin proved that the second-order theory of n successors is decidable.

Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory.

The use of second-order arithmetic also allows many techniques from recursion theory to be employed ; many results in reverse mathematics have corresponding results in computable analysis.

This book reprints much of Boolos's work on the rehabilitation of Frege, as well as a number of his papers on set theory, second-order logic and nonfirstorderizability, plural quantification, proof theory, and three short insightful papers on Gödel's Incompleteness Theorem.

One might attempt to reduce the second-order theory of the real numbers, with full second-order semantics, to the first-order theory in the following way.

Adler divides these second-order philosophical problems into two branches: one addressing the objects of thought, such as Being, Cause, Change, Infinity, Destiny, and Love ; the other addressing the subjects, or procedural domains, of thought, e. g. philosophy of religion, philosophy of history, philosophy of language, philosophy of science.

Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant, the same can be done for a second-order PDE at a given point.

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 ).

The stability condition stated above in terms of eigenvalues for the second-order case remains valid for the general n < sup > th </ sup >- order case: the equation is stable if and only if all eigenvalues of the characteristic equation are less than one in absolute value.

As mentioned in the introduction, the methods of renormalization have been applied to Statistical Physics, namely to the problems of the critical behaviour near second-order phase transitions, in particular at fictitious spatial dimensions just below the number of 4, where the above-mentioned methods could even be sharpened ( i. e., instead of " renormalizability " one gets " super-renormalizability "), which allowed extrapolation to the real spatial dimensionality for phase transitions, 3.

By plugging these expected runs scored and allowed into the pythagorean formula, one can generate second-order wins, the number of wins a team deserves based on the number of runs they should have scored and allowed given their component offensive and defensive statistics.

Boolos argued that if one reads the second-order variables in monadic second-order logic plurally, then second-order logic can be interpreted as having no ontological commitment to entities other than those over which the first-order variables range.

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.

Unlike first-order logic, which has only one standard semantics, there are two different semantics that are commonly used for second-order logic: standard semantics and Henkin semantics.

The second-order theory of the real numbers has only one model, however.

This follows from the classical theorem that there is only one Archimedean complete ordered field, along with the fact that all the axioms of an Archimedean complete ordered field are expressible in second-order logic.

This shows that the second-order theory of the real numbers cannot be reduced to a first-order theory, in the sense that the second-order theory of the real numbers has only one model but the corresponding first-order theory has many models.

A parametric equalizer, on the other hand, has one or more sections each of which implements a second-order filter function.

** In a second-order reference, one of Sandy Mitchell's Warhammer 40000 novels about Commissar Ciaphas Cain, a character largely based on Flashman, includes Commissar Tomas Beije, an old schoolmate of Cain, as a secondary character.

In general one distinguishes first-and second-order Stark effects.

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.

In the definition of a generalized BV algebra, one drops the second-order assumption for Δ.

Commercial applications of this approach involve combinations in which no two compounds ever share more than one well, to reduce the ( second-order ) possibility of interference between pairs of compounds being screened.

The tangent, curvature, and normal vector together describe the second-order behavior of a curve near a point.

Superfluidity in helium arises from the normal liquid by a second-order phase transition (" lambda transition ").

Because every formula has a prenex normal form, every formula in the language of second-order arithmetic is or for some.

This system is not expected to possess a normal second-order phase transition.

Stress ( mechanics ) | The stress tensor, a second-order tensor.

where is the Cauchy stress tensor, is the infinitesimal strain tensor, is the displacement vector, is the fourth-order stiffness tensor, is the body force per unit volume, is the mass density, is the divergence operator, represents the gradient operator and represents a transpose, represents the second derivative with respect to time, and is the inner product of two second-order tensors ( summation over repeated indices is implied ).

The strain tensor, similar in nature to the stress tensor -- both are symmetric second-order tensors --, is given in matrix form as