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The second-order logic without these restrictions is sometimes called full second-order logic to distinguish it from the monadic version. Monadic second-order logic is particularly used in the context of Courcelle's theorem, an algorithmic meta-theorem in graph theory. The MSO theory of the complete infinite binary tree is decidable.
Second-order arithmetic, an axiomatization allowing quantification of sets of numbers; Second-order differential equation, a differential equation in which the highest derivative is the second; Second-order logic, an extension of predicate logic; Second-order perturbation, in perturbation theory
The (full) second-order induction scheme consists of all instances of this axiom, over all second-order formulas. One particularly important instance of the induction scheme is when φ is the formula " n ∈ X {\displaystyle n\in X} " expressing the fact that n is a member of X ( X being a free set variable): in this case, the induction axiom ...
An example of second-order conditioning. In classical conditioning, second-order conditioning or higher-order conditioning is a form of learning in which a stimulus is first made meaningful or consequential for an organism through an initial step of learning, and then that stimulus is used as a basis for learning about some new stimulus. For ...
The order of a tensor is the sum of these two numbers. The order (also degree or rank) of a tensor is thus the sum of the orders of its arguments plus the order of the resulting tensor. This is also the dimensionality of the array of numbers needed to represent the tensor with respect to a specific basis, or equivalently, the number of indices ...
Priestley uses stationary up to order m if conditions similar to those given here for wide sense stationarity apply relating to moments up to order m. [ 3 ] [ 4 ] Thus wide sense stationarity would be equivalent to "stationary to order 2", which is different from the definition of second-order stationarity given here.
Second-order Taylor series approximation (in orange) of a function f (x,y) = e x ln(1 + y) around the origin. In order to compute a second-order Taylor series expansion around point (a, b) = (0, 0) of the function (,) = (+), one first computes all the necessary partial derivatives:
Second-order election is a term that appeared for the first time in Karlheinz Reif and Hermann Schmitt's "Nine second-order national elections – A conceptual framework for the analysis of European election results" article for the [European Journal of Political Research, in 1980]. [1]