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[1] [2] The order of the equation is the maximum time gap between any two indicated values of the variable vector. For example, = + is an example of a second-order matrix difference equation, in which x is an n × 1 vector of variables and A and B are n × n matrices. This equation is homogeneous because there is no vector constant term added ...
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 system ACA 0 is a conservative extension of first-order arithmetic (or first-order Peano axioms), defined as the basic axioms, plus the first-order induction axiom scheme (for all formulas φ involving no class variables at all, bound or otherwise), in the language of first-order arithmetic (which does not permit class variables at all).
[1] [2] The realization is called "minimal" because it describes the system with the minimum number of states. [2] The minimum number of state variables required to describe a system equals the order of the differential equation; [3] more state variables than the minimum can be defined. For example, a second order system can be defined by two ...
For an arbitrary system of ODEs, a set of solutions (), …, are said to be linearly-independent if: + … + = is satisfied only for = … = =.A second-order differential equation ¨ = (,, ˙) may be converted into a system of first order linear differential equations by defining = ˙, which gives us the first-order system:
A second-order stimulus is a type of visual stimulus employed in psychophysics where objects are distinguished from their backgrounds through variations in contrast or texture. In contrast, a stimulus distinguished by differences in luminance is termed a first-order stimulus.
In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order ...
My only reservation is that the first-order vs. second-order discussion seems to involve a fair amount of hand waving. it's also worth noting that the Peano axioms are generally (sop far as I, a nonspecialist, know) considered in a first order context, and they're not really expected to be categorical. Greg Woodhouse 15:50, 26 March 2007 (UTC)
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